Police authority A would inform police in city B. Police from city B would do the footwork, but someone from A would go there and hang around. If the two places were adjacent, then the police from A probably would go next door and do something.
That may be ideal, and typical, but it doesn't always happen that way.
A year or so ago police in one community in Michigan were tracking some suspected burglers. They didn't "chase" them, but did follow them as the burglers went from one community to another (several miles away).
At one point, the police were still watching them when the suspects pulled into the driveway of a home, and broke into the house.
The police then exited their vehicles and covered the front and back doors, waiting for the suspects to leave the house before aprehending them.
Unfortunately, one of the suspects found (stole) a pistol from inside the home and when confronted by an officer in the back yard, shot and killed the officer. He was then shot and killed by another officer. The other suspect was caught.
At no time did the police running the survellience on the suspects contact any of the other agencies as they followed the suspects through several (3 or 4) jurisdictions.
The first time the police in city where the shooting occurred heard of it was after the shooting when the "Officer down" calls and ambulance calls went out.
There was some controversy over this later and accusations that the originating police force mishandled the whole thing.
All the agencies involved were various city PD's btw, not Sheriff's or State Police.