Yeah, I think that gets tricky. There are plenty of counties or other governments that contract with private ambulance services to provide "public" 911/emergency response. Barnwell County SC contracts with Medshore for emergency ambulance service, for example. AMR (American Medical Response) and Rural/Metro are two of the big providers, at least in the southeast.
I think Cherokee County SC recently put out bids to companies to provide emergency medical service after their contract with their local hospital - Upstate Carolina, which was later bought by Mary Black Hospital, which was recently purchased by Spartanburg Regional Medical Center - ran out and was not renewed. In fact, here's a news article from August 2019 saying that Lifeguard Ambulance will soon be providing 911 emergency medical service in Cherokee Co.
Cherokee County Council approves new ambulance service I think that Lifeguard might be the service using talkgroup 51733 in the Midlands, that I posted about last week in the OpenHmz.com thread regarding Sumter County.
Many big hospital systems (quasi-private entities now) also provide non-emergency transports that compete with for-profit ambulance services. In the Upstate I see Ambu-star, Transmed, Family Medial and numerous others along with ambulances marked as operated by Spartanburg Regional, though they aren't emergency response ambulances.
The ambulance service (aka Medicare) is obviously a lucrative business to be in.