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 inst…