You are correct. West Caldwell PD currently dispatches. When you hear Signal 9-1 that means Duty Crew. Signal 9 is a rig back up crew requested. 9-2 is a second rig. 9-3 is a call for the Rescue Truck. ("You may hear 9-2 & a 9" meaning back up crew and no one assigned)
Dispatchers are assigned numbers and are on call at home. With one exception. That being when no volunteer dispatcher is on for all the responders to call for the dispatch info and then they are told a specific dispatcher number which means call the desk. They have conducted these ops for many, many years. Cencom fits into the equation by way of providing Pre Arrival Instructions to the various West Essex Municipalities. So you would call 9-1-1 and the PSAP dispatcher would take some basic info, then transfer the call when approriate to Cencom for Pre Arrival Instructions by a EMD @ Cencom.
It also allows for automatic dispatch of the medics.