Not necessarily. I think in CA it's still on the books that a "driver" doesn't need formal EMS training. that being said, I think they aren't supposed to be touching patients either!
Same here in Illiois, the service I worked at in the early 90's had a guy that failed his EMT test 5 times, but they let him work as a "driver". He could NOT legally touch a patient, but we had to keep reminding him of that almost daily. Every chance he got when he thought you were to busy to watch him, he would start trying to "treat" people.