What is the name of your state (only U.S. law)? Ohio
I know this is going to depend on which police department we are talking about here. I believe state troopers have the right to any public roads like highways. Plus they can be brought in if requested by a local police agency.
However, can a city or county police officer sit in one county (the county they do not belong.) and then make an arrest in the country in which they do belong for a stop light violation? I am sure they can do the reverse but would think they would not be allowed to sit in another county.