As noted,, it's basically about jurisdiction.
If you live within a city limit area,, it's usually a city LEO that handles things.
If you live in the County,, then the Sheriff Dept has the authority.
And of course,, on the state & federal highways,, it's the State Highway patrol.
Of course,, there are agreements ( read, monetary compensation) where a city may not have a police dept, or similar situations.
And depending upon what laws are broken,, it can bring in other LEO agencies.
As for where I live.
My property actually has the County line running through it. So, I'm on the edge. Plus,, a city Dept is actually the closest LEO agency to my home. As such,, I keep up with (2) County Sheriff's, and a City Chief of Police. Overall,, I'm pretty well satisfied with all of them. If we need something,, they have all be very good about responding.
But if it were a very serious thing,, all 3 would likely show up.
And my youngest son is LEO in a neighboring County. I keep up with many from his Dept. He has a good Sheriff & lots of good deputies.
Then, another nearby County,, well, the Sheriff is not very good at all. And the main city in that County,, just as bad. The City has gone through several Chiefs in a very short period of time. Lots of problems with the local politics, local attitudes towards LEO's, and such. Many of the patrol officers try really hard,, but struggle sometimes due to the lack of support from the politicians, and others.
That County? Buncombe. the City? Asheville NC.
Sadly,, long ago,, Buncombe County suffered from a very corrupt Sheriff. Once he was ousted,, the next Sheriff took on a severe crap-storm. Yet,, he not only turned everything around,, it became a shining example of how a professional Sheriff Dept should be run. That gentleman did one heck of a job. Once he retired,, things went downhill again. Not quite to the level it had been prior to the GOOD Sheriff,, but it's bad currently.
But that top level, excellent Sheriff is trying to get on the local ballot for the upcoming election, as a County Commissioner. A LOT of us are hoping he can get elected. He was at my FoNRA event last Friday night,, and just the same great guy he's always been. I wish I could vote in his County.
But in general,, we do have a pretty good bunch of Sheriff Dept's all around our area,, and I'm very happy for that. Would I change how they are elected, or wish for a union? HECK NO!!!!!!