However both police and firemen already deal with those issues, especially at the edges of cities. E.g. where I live, at the edge of two, police from both regularly handle incidents.
Firemen are even more heterogeneous, in that they have arrangements for big fires and catastrophes like tornadoes where adjacent units come to help. After my city got hit by an EF-5 tornado, as I was exiting my trashed apartment complex, there were fire units varying from a small Indian tribe that's right across a state border from us (and the city e.g. treats the sewage of their casino) to one 40 miles away. Next day, the markings on my apartment door were from a fire unit the came from 160 miles away.
What is needed is something like unity of command/hierarchy. Having two or more units responsible for the same territory leads to the sorts of problems you cite, e.g. a friend told me the NYC city police and subway police once got into a big firefight. This would have been in the '70s or earlier, now they've been unified.
