Testing Atezolizumab Alone or Atezolizumab Plus Bevacizumab in People With Advanced Alveolar Soft Part Sarcoma

Inclusion Criteria:

* Patients must have histologically or cytologically confirmed alveolar soft part sarcoma that is not curable by surgery. Diagnosis of malignancy must be confirmed by the department of pathology at the institution where the patient is enrolled prior to patient enrollment. To be eligible for atezolizumab in combination with bevacizumab, the patient must have atezolizumab-refractory/resistant disease that has progressed (definitive clinical progression or confirmed progressive disease \[iCPD\]) on prior 10005 atezolizumab monotherapy
* Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) by chest x-ray or as \>= 10 mm (\>= 1 cm) with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam

  * Note: Once the primary endpoint of the trial has been met, pediatric patients may enroll with evaluable non-measurable disease and will not be required to have measurable disease. Evaluable non-measurable disease is that which is not measurable by RECIST1.1 but can be evaluated by imaging (e.g., CT, bone scan, or ultrasound) or other methods
* Patients with newly diagnosed, unresectable, metastatic and measurable ASPS will also be eligible for atezolizumab monotherapy if they show clinical evidence of disease progression (including history and increasing physical symptoms); on-study documentation will include a physician's rationale that supports evidence of clinical disease progression (i.e., increasing tumor pain)
* To be eligible for atezolizumab monotherapy, subjects must be \>= 2 years of age. To be eligible for atezolizumab in combination with bevacizumab, subjects must be \>= 18 years of age
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky or Lansky \>= 70%)
* Life expectancy of greater than 3 months
* Absolute neutrophil count \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* Hemoglobin \>= 8 g/dL
* Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (however, patients with known Gilbert disease who have serum bilirubin level =\< 3 x ULN may be enrolled)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 3 x ULN (AST and/or ALT =\< 5 x ULN for patients with liver involvement)
* Alkaline phosphatase =\< 2.5 x ULN (=\< 5 x ULN for patients with documented liver involvement or bone metastases)
* Creatinine clearance for adult patients (\>= 18 years of age): \>= 30 mL/min/1.73 m\^2 by Cockcroft-Gault; for pediatric patients (\< 18 years of age), a serum creatinine based on age and sex as follows:

  * Age 2 to \< 6 years, maximum serum creatinine (mg/dL) male 0.8, female 0.8;
  * Age 6 to \< 10 years, maximum serum creatinine (mg/dL) male 1, female 1;
  * Age 10 to \< 13 years, maximum serum creatinine (mg/dL) male 1.2, female 1.2;
  * Age 13 to \< 16 years, maximum serum creatinine (mg/dL) male 1.5, female 1.4;
  * Age 16 to \< 18 years, maximum serum creatinine (mg/dL) male 1.7, female 1.4
* Administration of atezolizumab or bevacizumab may have an adverse effect on pregnancy and poses a risk to the human fetus, including embryo-lethality. Female patients of child-bearing potential and male patients must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 5 months (150 days) after the last dose of atezolizumab or 6 months (180 days) after the last dose of bevacizumab. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document or a parent/guardian able to do the same
* Willingness to provide biopsy samples for research purposes (patients \>= 18 years of age only). At the principal investigator (PI's) discretion, archival tissue that was collected and preserved may be used in lieu of a baseline biopsy, provided no anti-cancer agents or immunotherapies were received since collection of the archival tissue
* Patients who have received live attenuated vaccines within 30 days of the first dose of trial treatment are eligible at the discretion of the investigator. All seasonal influenza vaccines and vaccines intended to prevent SARS-CoV-2 and coronavirus disease 2019 (COVID-19) are allowed
* For patients not receiving therapeutic anticoagulation: international normalized ratio (INR) or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN for bevacizumab patients
* For patients receiving therapeutic anticoagulation: stable anticoagulant regimen for bevacizumab patients

Exclusion Criteria:

* Any prior therapy must have been completed \>= 4 weeks or, if known, \>= 5 half-lives of the prior agent (whichever is shorter) prior to enrollment on protocol (minimum of 1 week between prior therapy and study enrollment), and the participant must have recovered to eligibility levels from prior toxicity. Note: For patients being treated on the atezolizumab plus bevacizumab arm, there is no washout requirement for prior anti-PD-1 or anti-PD-L1 agents.

  * Patients should be at least 6 weeks out from nitrosoureas and mitomycin C. Prior definitive radiation should have been completed \>= 4 weeks or palliative radiation should have been completed \>= 2 weeks prior to study enrollment or crossover and all associated toxicities resolved to eligibility levels (patients on study may be eligible for palliative radiotherapy to non-targeted lesions after 2 cycles of therapy at the principal investigator \[PI\]'s discretion). Patients who have had prior monoclonal antibody therapy must have completed that therapy \>= 6 weeks (or 3 half-lives of the antibody, whichever is shorter) prior to enrollment on protocol (minimum of 1 week between prior therapy and study enrollment). A patient who has received a cumulative dose of \> 350 mg/m\^2 of anthracycline (regardless of cardioprotectant) may only be enrolled if their ejection fraction measured by an echocardiogram is within normal institutional limits
* Prior treatment with anti-PD-1 or anti-PD-L1 therapeutic antibody, or pathway-targeting agents

  * Patients who have received prior treatment with anti-CTLA-4 may be enrolled on the atezolizumab monotherapy arm, provided the following requirements and all other selection criteria are met:

    * Minimum of 12 weeks from the first dose of anti-CTLA-4 and \> 6 weeks from the last dose
    * No history of severe immune-related adverse effects from anti-CTLA-4 (NCI Common Terminology Criteria for Adverse Events \[CTCAE\] grade 3 and 4)
  * Patients who have progressed on P10005 atezolizumab monotherapy may be eligible for the atezolizumab plus bevacizumab arm, provided the following requirements, and all other selection criteria, are met:

    * No prior disease progression on an immune checkpoint inhibitor plus tyrosine kinase inhibitor combination therapy
    * Any prior toxicities have resolved to eligibility levels
    * No contraindications
* Treatment with any other investigational agent within 4 weeks (or within five half-lives of the investigational product, whichever is shorter) prior to cycle 1, day 1 (minimum of 1 week between prior therapy and study enrollment); patients must be \>= 2 weeks since any investigational agent administered as part of a phase 0 study (also referred to as an "early phase I study" or "pre-phase I study" where a sub-therapeutic dose of drug is administered) at the coordinating center PI's discretion, and should have recovered to eligibility levels from any toxicities
* Treatment with systemic immunostimulatory agents (including, but not limited to, interferon-alpha or interleukin-2 \[aldesleukin\]) within 6 weeks prior to cycle 1, day 1
* Treatment with systemic immunosuppressive medications (including, but not limited to, prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[anti-TNF\] agents) within 2 weeks prior to cycle 1, day 1

  * Patients who have received acute, low dose, systemic immunosuppressant medications (e.g., a one-time dose of dexamethasone for nausea) may be enrolled
  * The use of inhaled corticosteroids and systemic mineralocorticoids (e.g., fludrocortisone) for patients with orthostatic hypotension or adrenocortical insufficiency is allowed
* Patients taking bisphosphonate therapy for symptomatic hypercalcemia; use of bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is allowed
* Patients with known primary central nervous system (CNS) malignancy or symptomatic CNS metastases are excluded, with the following exceptions:

  * Patients with asymptomatic untreated CNS disease may be enrolled, provided all of the following criteria are met:

    * Evaluable or measurable disease outside the CNS
    * No metastases to brain stem, midbrain, pons, medulla, or cerebellum
    * No history of intracranial hemorrhage or spinal cord hemorrhage
    * No ongoing requirement for dexamethasone for CNS disease; patients on a stable dose of anticonvulsants are permitted
    * No neurosurgical resection or brain biopsy within 28 days prior to cycle 1, day 1
  * Patients with asymptomatic treated CNS metastases may be enrolled, provided all the criteria listed above are met as well as the following:

    * Radiographic demonstration of improvement upon the completion of CNS-directed therapy and no evidence of interim progression between the completion of CNS-directed therapy and radiographic screening for the current study
    * No stereotactic radiation or whole-brain radiation within 28 days prior to cycle 1, day 1
    * Screening CNS radiographic study \>= 4 weeks from completion of radiotherapy and \>= 2 weeks from discontinuation of corticosteroids
* Known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies
* History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies (i.e., antibodies with generic names ending in "ximab" or "zumab", respectively) or fusion proteins
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study because atezolizumab is an investigational agent with the unknown potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with atezolizumab, breastfeeding should be discontinued if the mother is treated with atezolizumab
* Human immunodeficiency virus (HIV)-positive patients may be on combination antiretroviral therapy but must have a CD4 count \> 350 cells/mm\^3 with an undetectable viral load
* Patients on supraphysiologic doses of steroids or use of such within the previous six weeks
* Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease.

  * Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible
  * Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA)
* History or risk of autoimmune disease, including, but not limited to, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Bell's palsy, Guillain-Barre syndrome, multiple sclerosis, autoimmune thyroid disease, vasculitis, or glomerulonephritis

  * Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone may be eligible
  * Patients with controlled type 1 diabetes mellitus on a stable insulin regimen may be eligible
  * Patients with eczema, psoriasis, lichen simplex chronicus or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided that they meet the following conditions:

    * Patients with psoriasis must have a baseline ophthalmologic exam to rule out ocular manifestations
    * Rash must cover less than 10% of body surface area (BSA)
    * Disease is well controlled at baseline and only requiring low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%)
    * No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids)
* History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan; history of radiation pneumonitis in the radiation field (fibrosis) is permitted
* Patients with active tuberculosis (TB) are excluded
* Severe infections within 4 weeks prior to cycle 1, day 1, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia
* Signs or symptoms of infection within 2 weeks prior to cycle 1, day 1
* Received oral or intravenous (IV) antibiotics within 2 weeks prior to cycle 1, day 1; patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract infection or chronic obstructive pulmonary disease) are eligible
* Major surgical procedure within 28 days prior to cycle 1, day 1 or anticipation of need for a major surgical procedure during the course of the study. Note: Patients with a surgical procedure (excluding biopsy procedures) within 28 days prior to cycle 1 day 1, or any unhealed surgical wound, are not eligible for the atezolizumab + bevacizumab combination arm
* Administration of a live, attenuated vaccine within 4 weeks before cycle 1, day 1 or anticipation that such a live, attenuated vaccine will be required during the study and up to 5 months after the last dose of atezolizumab

  * Influenza vaccination should be given during influenza season only (approximately October to March); patients must not receive live, attenuated influenza vaccine within 4 weeks prior to cycle 1, day 1 or at any time during the study
* Additional exclusion criteria include:

  * History of leptomeningeal disease
  * Uncontrolled tumor-related pain
  * Patients requiring pain medication must be on a stable regimen at study entry.
  * Symptomatic lesions (e.g., bone metastases or metastases causing nerve impingement) amenable to palliative radiotherapy should be treated prior to enrollment. Patients should be recovered from the effects of radiation. There is no required minimum recovery period.
  * Asymptomatic metastatic lesions that would likely cause functional deficits or intractable pain with further growth (e.g., epidural metastasis that is not currently associated with spinal cord compression) should be considered for loco-regional therapy if appropriate prior to enrollment.
  * Uncontrolled pleural effusion, pericardial effusion, or ascites requiring recurrent drainage procedures (once monthly or more frequently)
  * Patients with indwelling catheters (e.g., PleurX) are allowed.
  * Uncontrolled or symptomatic hypercalcemia (ionized calcium \> 1.5 mmol/L, calcium \> 12 mg/dL or corrected serum calcium \> ULN)
  * History of malignancy within 3 years prior to screening, with the exception of malignancies with a negligible risk of metastasis or death (e.g., 5-year OS rate \> 90%), such as adequately treated carcinoma in situ of the cervix, non melanoma skin carcinoma, localized prostate cancer, ductal carcinoma in situ, or stage I uterine cancer
  * Prior allogeneic stem cell or solid organ transplantation
  * Any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of an investigational drug, may affect the interpretation of the results, or may render the patient at high risk from treatment complications
* BEVACIZUMAB-SPECIFIC EXCLUSION CRITERIA
* Age \< 18 years
* Prior disease progression on an immune checkpoint inhibitor plus tyrosine kinase inhibitor combination therapy
* Major surgery within 6 weeks of enrollment. Procedures (e.g. port placement, endoscopy with intervention) within 7 days of enrollment (excluding biopsy procedures)
* Chronic daily treatment with a non-steroidal anti-inflammatory drug (NSAID), clopidogrel, dypiridamole, or aspirin therapy \> 81 mg/day. Treatment with any such agents within 7 days prior to cycle 1 day 1
* 2+ protein on urinalysis, followed by 24-hour protein of \> 1 g
* Patients with a history of bleeding varices within 1 year of enrollment
* Thromboembolic event within 6 months of enrollment (including cerebrovascular accident \[CVA\] and myocardial infarction (MI)
* Evidence of bleeding diathesis or significant coagulopathy (with or without current therapeutic anticoagulation)
* International normalized ratio (INR) \>= 1.5 (or \> therapeutic range if patient is receiving warfarin). Partial thromboplastin time (PTT) \>= 1.5 x ULN
* History of hemoptysis (\> 1/2 teaspoon of bright red blood per episode) within 1 month of enrollment
* Serious, non-healing wound, active ulcer, or untreated bone fracture
* Congestive heart failure, baseline left ventricular ejection fraction (LVEF) \< 50%, transmural myocardial infarction, uncontrolled hypertension (defined as systolic blood pressure \> 140 mmHg and/or diastolic blood pressure \> 90 mmHg in patients \>= 18 years of age), angina pectoris requiring medication, clinically significant valvular disease, high-risk arrhythmia within 12 months of enrollment. Prior history of hypertensive crisis or hypertensive encephalopathy
* History of abdominal fistula or gastrointestinal perforation within 6 months prior to enrollment
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to bevacizumab
* For patients not receiving therapeutic anticoagulation: INR or aPTT =\< 1.5 x; for patients receiving therapeutic anticoagulation: stable anticoagulant regimen
* Significant vascular disease (e.g., aortic aneurysm requiring surgical repair or recent arterial thrombosis) within 6 months prior to randomization
* Core biopsy or other minor surgical procedure, excluding placement of a vascular access device, within 7 days prior to initiation of study treatment. Placement of a vascular access device should be at least 2 days prior to initiation of study treatment
* Active infection requiring IV antibiotics at the time of initiation of study treatment
* Current or recent (\< 10 days prior to initiation of study treatment) use of aspirin (\> 325 mg/day), or clopidogrel (\> 75 mg/day)
* Note: The use of full-dose oral or parenteral anticoagulants for therapeutic purpose is permitted as long as the INR and/or aPTT is within therapeutic limits (according to institution standards) within 7 days prior to initiation of study treatment and the patient has been on a stable dose of anticoagulants for \>= 2 weeks prior to in