Yes stuck vehicle was just first thing that came into my head, thouh actually I beleive one of the Norfolk Response groups is the first line of recovery for their local ambulance service for anything that has left the road, be it only in a ditch, commercial recovery unless the real specialists aren't really that good at getting stuff out of fields and ditches whereas the 4x4 community do it every weekend for fun (well some do any way), specialists tend to not be available on a do it now basis as they may well be doing another job riht then. Anyway I digress. But no we wouldn't go out to joe public stuck somewhere unless requested to by Police etc, we did remove a couple of vehicles off scarborough marine drive during the storm surge for instance.
A call is handled by a "Contoller", (I'm one), these will designate a team leader for on the ground but in many cases this will be supperflous really as Responders will often end up working direct under who ever called. For instance I went to Grimsby during the floods of 2007 and never saw another responder till the very end of the day when I was releive. I turned up at the designated Police station on from then on I was at the disposal of whoever wanted me/I was asigned to. Ended up patrolling flooded estates on a hearts and minds type excercise and then acting as a road block to prevent the local 4x4s causing more damage by driving through the floods causing big waves.
The same happened in the recent snow incidents in Lincs and East Yorks, I was controller and called the 13 responders out, gave them details etc but once they got to scene beyond reporting in to me they were there and then they were leaving the were in the control of the officers on the ground. I then waited till everyone was home safe and sound before standing down myself.
The controller (possibly in concert with other controllers) makes the call as to whether we should deploy, if its a Cat 1 we almost certainly would, in other circumstance we might not or might elect to clear it through a Cat1 agency first. For instance we recovered a vehicle from the Humber foreshore which was below the high water mark one time.
Other callouts that are more than Public Assistance or Good Samaritan type acts but not a major incidents have included, assisting Vets and British Marine Divers Assc to access Spurn Point and a stricken seal. Recovering Humber Rescue when they got their RIB on trailer stuck on Humber Foreshore etc.