Atezolizumab With or Without Cobimetinib in Treating Patients With Metastatic Bile Duct Cancer That Cannot Be Removed by Surgery or Gallbladder Cancer

Inclusion Criteria: * Pathologically confirmed metastatic or unresectable cholangiocarcinoma or gallbladder carcinoma (GBC), having received at least 1 prior line of systemic therapy, and received no more than 2 prior lines of therapy in the metastatic setting (disease recurrence =\< 6 months from the last dose of adjuvant therapy in resected patients will be considered the first line of therapy) * Includes intrahepatic cholangiocarcinoma (IHC), extrahepatic cholangiocarcinoma (EHC), and gallbladder carcinoma (GBC), but not ampulla of vater cancers * 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) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral CT scan, MRI, or calipers by clinical exam; assessment must be completed within 4 weeks of randomization * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of cobimetinib in combination with atezolizumab in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 80%) * Life expectancy of greater than 2 months * Leukocytes \>= 2,500/mcL (within 2 weeks of randomization) * Absolute neutrophil count \>= 1,500/mcL (within 2 weeks of randomization) * Platelets \>= 75,000/mcL (within 2 weeks …