Interesting.
My ambulance is county
Which county?? You might need to check again. Some places/providers, like LA County Fire Dept (and parts of Orange County Fire Authority), subcontract private ambulance companies to do the actual transports. Those companies function like I mentioned.
If it truly is municipal-based, I would think that taxes would handle the charge, and it should be free. Then again, if they don't get tax money, they would charge.
California has a complicated system, huh?