All 911 dispatching is done by the FDNY. We also control pretty much all of EMS in the city except for nonemergency transports. Oversight of everything is done by NYC Resmco which falls under the State Dept of Health.
Every 911 unit, Fire and Private have MDT's and login at the start of each tour. The Dispatcher sees everyone as them same and the jobs go out. There are a few exceptions and there are allways games that are played.
The volunteer ambulance also have access to the FDNY but it's somewhat limited and could use an overhaul.