Testing the Combination of Anti-cancer Drugs Atezolizumab and Tiragolumab in People With Advanced Stage Rare Cancers, RARE3 Trial

Inclusion Criteria:

* Patients must have histologically confirmed rare solid tumors that have progressed on standard therapy or for whom there is no standard of care therapy

  * Patients must not be eligible for a higher priority study that would be feasible for them to enroll in, such as a disease specific study of phase 2 or higher or a randomized study. Specifically, patients who are eligible for the PEP-CTN pediatric trial of atezolizumab and tiragolumab in children, adolescents, and young adults with SMARCB1- or SMARCA4-deficient tumors should be excluded
* Patients must have measurable disease as defined by RECIST v1.1, with 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 CT scan, MRI, or calipers by clinical exam)
* Patients must have a tumor site amenable to biopsy
* Age \>= 18 years. Because biopsies are mandatory on this trial, patients \< 18 years of age are excluded
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Absolute neutrophil count \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* International normalized ratio (INR) or activated partial thromboplastin time (aPTT) =\< 1.5 institutional upper limit of normal (ULN)

  * Patients who receive therapeutic anticoagulation therapy should be on a stable dose
* Total bilirubin =\< 1.5 x institutional ULN (however, patients with known Gilbert disease who have serum bilirubin level of up to 3 mg/dl may be enrolled)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic-pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional ULN (AST and/or ALT =\< 5 x ULN for patients with liver involvement)
* Creatinine =\< 1.5 x institutional ULN OR creatinine clearance levels \>= 30 mL/min/1.73 m\^2 are permitted as the study agents are not secreted by the kidney
* Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial
* For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated
* Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load
* Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression for more than \>= 1 month after treatment of the brain metastases
* Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy
* Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
* Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better
* Willingness to provide biopsy samples for research purposes
* Administration of atezolizumab and tiragolumab 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 study agent. 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

Exclusion Criteria:

* Patients who have had prior monoclonal antibody therapy must have completed that therapy \>= 5 weeks (or 3 half-lives of the antibody, whichever is shorter) prior to start of treatment (minimum of 1 week between prior therapy and study enrollment)
* Patients must have recovered from clinically-significant adverse events of their most recent cancer immunotherapy to grade 1 or less, (with the exception of alopecia and lymphopenia)
* Patients who are receiving any other investigational agents
* Prior anti-TIGIT therapy is not allowed. However, other prior immune checkpoint inhibitor therapy is permitted
* 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, not resolved by pre-medication or steroids, leading to subsequent treatment cessation. Patients with a history of allergic reaction to chimeric or humanized antibodies for which symptoms never recurred after subsequent re-challenge may be considered after careful medical history review
* Treatment with systemic immunosuppressive medications (including, but not limited to, prednisone \[\> 10 mg/day\], 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 with uncontrolled intercurrent illness, that would limit compliance with study requirements
* Pregnant women are excluded from this study because atezolizumab and tiragolumab are investigational agents with 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, and because it is not known if tiragolumab can be excreted in human milk, breastfeeding should be discontinued if the mother is treated with atezolizumab
* 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 autoimmune hyperthyroid disease not requiring immunosuppressive treatment 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)
* 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
* Patients who have undergone major surgical procedures prior to Cycle 1, Day 1 who have not recovered to ECOG performance status =\< 2 (Karnofsky \>= 60%)
* 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