There are minor differences from state to state, generally speaking. But unlike Germany which is roughly the size of a medium-sized state here...well, with the addition of the east maybe a state and a half, the differences from one region to another in the US are very pronounced. There are minor cultural differences from Hesse to Bayern, nicht war? Laws in Arizona about car travel across the desert and the implementation of a sheriff's posse to search for lost souls would not make sense in a place like Massachusetts or even Dade county in Florida.
A police department is generally a city organization. Most states have a state police, too, sometimes called Highway Patrol, sometimes Department of Public Safety, but still a police force with jurisdiction over the whole state. City police have jurisdiction only in their city limits. The Sheriff is usually the law enforcement arm of the county, with specific duties in the county areas and support for town or city cops, too.
The way it is supposed to work is that local law enforcement have the authority to manage their own local areas. State and Federal legislators will pass bills that are designed to integrate policies and laws. For example. I live in El Paso County in Colorado. A few years ago we had a real smart Sheriff named John Anderson. He relaxed the requirements for citizens to get a concealed weapons permit. By state law, once a sheriff issues the permit, it is supposed to be good throughout the state. This caused a lot of controversy from county to county because not everyone in this colorful state has a brain. But after a few years of watching the crime statistics drop in El Paso County, the state began to work on a bill to put the whole state under the same rules that we have here to obtain a carry permit.
Hope that helps clear it up, and I am sure that other folks will have a different idea about how things should work, but there it is.
