I was once naive enough to think that that was the way all fire departments work. It's been a while since I thought this way but I still get angry when I see the way some cities/departments provide emergency care.
Let me give you an example of what I think is a better way of doing things (maybe not the best but better). I grew up in Southern California. There are many largish fire departments (between 6 and 10 stations each) in the area that I called home. These departments all started using one dispatch center, reducing response times because you don't have to transfer the call to another dispatch center.
Another way they improved service was to drop all borders. Each unit has a GPS tracker. When a call comes in the computer tells the dispatcher which is the closest unit to the call, regardless of the jurisdiction. The cities and fire departments (and I would think PD as well but I'm not sure) mutually decided that getting help to the citizen in need was of more importance than who provided the help.
We'd love to help you but you're on the wrong side of the dotted line on the map.
This is not always the case. There are several departments with which we share borders. For various reasons (in some cases noses of some politicians have been bent out of shape, another case, the members of the FD act as if they are better than us) these smaller departments feel that they do not need to have the closest unit respond to an emergency instead electing to have their own unit respond.
In one of the cities my department has a station within their city limits. Yet, if the person that lives next door to our station calls 911, he will have the engine from almost 2 miles away respond. The captain on that engine may or may not request that we be dispatched since we're closer.
One day, while working overtime at that station, we had one of the residents from a half block down stop by with his son for a station visit. The dad was telling his son that if they had an emergency we would come help. I stopped him and explained that we would probably not get dispatched to the call even though we are the closest unit.
It's frustrating and childish to me. I would love to be able to sit the people in charge down and ask them if they think they are really trying to provide better care to the people. Stop getting into pissing contests like you're 13 and start doing what's right for your communities.
Progress is coming but it's slow.