This would be Metro EMS (where I work) - we have a RESCUE unit and then AMBULANCE unit.
RESCUE unit goes out to specialized calls - think of Chicago Fire (TV Series) they have a specialized rescue team that Kelly Severide is on... pretty much the same thing.
Ambulance side generally gets dispatched and if we need assistance, we call for rescue to come back us up unless FIRE & RESCUE is on the scene and they can assist, then they do.
It's interesting to note that FIRE & RESCUE don't always respond to MEDICAL CALLS anymore... unless our control room requests that they do (probably because we busy / don't have ambulances available) but if it's TRAUMA, then they generally go, especially if MVA's and such.
Calls and vehicles dispatched to those calls also depend on what information is given.
Now, "WE" (Metro EMS) have a 'agreement' (if you want to call it that) where our control room gives calls to the private services.
I'm not entirely sure of the details, but for example, if there is an MVA (motor vehicle accident) the call will most probably be given to ER24 or Netcare 911 - if they are busy (no ambulances available) then they will dispatch one of our ambulances.
Private services claim from RAF for those type of calls, especially if YOU don't have medical aid.