Short answer: Medicare pays first for Medicare-covered services, and Medicaid pays last, after Medicare and any other insurance. Prescription drugs come through a Medicare Part D plan, not Medicaid.
Medicare.gov: people with Medicare and full-benefit Medicaid are dually eligible. Medicare pays first for Medicare-covered services; Medicaid pays last, after Medicare and any other health insurance the person has (such as a retiree plan).
Drugs work differently from what many families expect. A dual eligible’s prescriptions are covered by Medicare, and the person is automatically enrolled in a Medicare drug plan that covers them instead of Medicaid; Medicaid may still cover a drug Medicare doesn’t in certain situations. Duals also get Extra Help automatically.
Medicaid can also cover things Medicare generally doesn’t, such as nursing home care and personal care services. A dual eligible can still choose Original Medicare or Medicare Advantage, including plans designed for duals.