It used to be more common, but Prop 2 1/2 resulted in a lot of police operated ambulance services being turned over either to private services or the FD. Some FD (Brockton being one) got out at the same time. Quincy, Braintree, Dedham, Waltham and I think Fairhaven ran police based ambulances at one time.
Billerica is a pretty good service. Most, but not all of their medics are non sworn personnel. At least they were when we worked with them to get municipal EMS employees into Group 4 retirement.
Interestingly (to me) Fall River FD used to have non sworn personnel on the ambulances, but a couple of years ago split EMS off into a separate agency.