• Patients must be kidney transplant recipients with a functioning allograft who do not currently require dialysis

• Patient's age must be \>= 18 years. Because no dosing or adverse event (AE) data are currently available on the use of nivolumab and ipilimumab in kidney transplant recipients \<18 years of age, children are excluded from this study, but may be eligible for future pediatric trials

• Patients must have histologically or cytologically confirmed non-uveal melanoma, basal cell carcinoma, Merkel cell carcinoma, or cutaneous squamous cell carcinoma for which standard non-immunological medical, surgical, or radiation therapy would be insufficient (i.e., patients who are not surgical candidates). Patients with cutaneous squamous cell carcinoma or Merkel cell carcinoma may enroll without prior medical therapy (e.g., cetuximab or chemotherapy respectively). Non-immunologic standard therapies that patients must have received, refused or for which patients were ineligible include:

‣ For patients with BRAF-mutant melanoma, prior therapies include BRAF/MEK inhibitors

⁃ For patients with Basal cell carcinoma, prior therapies include hedgehog pathway inhibitors

• Patients must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria, i.e., 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 by chest x-ray or as \>= 10 mm with CT scan, magnetic resonance imaging (MRI), or calipers by clinical exam is preferred, but not required

• Patients must have Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) performance status criteria

• Leukocytes \>= 2,000/mcL

• Absolute neutrophil count \>= 1,500/mcL

• Platelets \>= 50,000/mcL

• Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN)

• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional ULN

• Serum creatinine =\< 3 x ULN

• dd-cfDNA =\< 1.0% and =\< 61% increase

• The effects of nivolumab and ipilimumab on the developing human fetus are unknown. For this reason, and because other therapeutic agents used in this trial are known to be teratogenic, women of childbearing potential (WOCBP) receiving nivolumab must continue contraception for a period of 5 months after the last dose of nivolumab. Women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile) as well as azoospermic men do not require contraception.

∙ WOCBP must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of beta-human chorionic gonadotropin \[B-HCG\]) during the screening period. Follow-up evaluations will include interval sexual/menstrual histories as needed.

∙ Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she (or the participating partner) should inform the treating physician immediately.

∙ WOCBP is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. Women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL to be considered postmenopausal.

• Human immunodeficiency virus (HIV)-infected patients will be eligible for this trial if they are on effective antiretroviral regimens utilizing non-CYP-interactive agents and have an undetectable viral load. If there is evidence of chronic hepatitis B virus (HBV) infection, HBV viral load must be undetectable on suppressive therapy, if indicated. If there is history of hepatitis C virus (HCV) infection, the patient must have been treated and have undetectable HCV viral load.

• Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants