There’s a lot more people than one might realize who are eligible for both Medicare and Medicaid. When someone is enrolled in both, the Medicare insurance coverage is primary and Medicaid is secondary.
So how it would work when you go to the doctor is you’d have to give them both insurance policy numbers. Every provider will ask you for “COB” information (coordination of benefits), which basically means that they’re asking if you have more than 1 insurance plan for medical coverage. If you’re enrolled in both Medicare and Medicaid, then the answer to that type of question would be yes. 
Sometimes based on which state you live in, what Medicare will pay/cover can be slightly different than what Medicaid will pay for/cover. The payment of the claim is all handled by the insurance companies.
Depending on the laws in different states, you can choose to be enrolled by the same insurance company for both your Medicare and Medicaid plans. Or you can be enrolled with a Medicare plan with one company but be enrolled in Medicaid with a different company.
Sometimes, insurance companies where you can enroll in both the Medicare and Medicaid coverage at the same company will have special ‘dual-eligible’ member programs for care management, coordination of care, and other programs for dual-eligible members with specific categories of health conditions.