THE SHORT ANSWERState law creates most local police powers and criminal rules. A city, county, or tribal government funds and governs its agency under local law; a chief or sheriff runs operations; prosecutors make charging decisions; courts decide cases; federal agencies handle federal crimes and can enforce federal civil-rights law.
Criminal law, budgets, department policy, operations, prosecution, oversight, and civil-rights enforcement belong to different actors.
Authority map
Follow the decision, not the title.
Each institution has a different verb. Writing a rule, funding it, applying it, administering a service, and reviewing a dispute are not the same power.
What changes by place
The exact controlling text still wins.
Strong-mayor and council-manager cities, independently elected sheriffs, consolidated governments, tribal sovereignty, state police, special police agencies, prosecutors, unions, consent decrees, and civilian oversight bodies divide authority differently.