This depends on the state, and the truth is a lot of places don't have specific laws relating to this since there are so few physicians working in the field. It's important to note the different between responding as medical control physician, where the medics are already operating under your license and you are already liable, and working as a medic. It gets tricky when you say "I have an MD, but I'm just working as an EMT/medic today." I know that medics work as EMTs a fair amount, they may be on a BLS ambulance for the day and they operate at the BLS level (since that ambulance won't be carrying ALS equipment.
Since the legal aspects are so murky, it would be wise to have some additional, personal liability/ malpractice insurance since if you are an attending the hospital that you work for probably doesn't cover you for working off duty, and the ambulance service may not cover you if you are held to a higher standard than a medic in court.
Note that none of this applies to residents doing ride alongs as part of their residency. You are covered by your residency malpractice as part of that (though in truth you aren't really going to be doing independent decision making, you are assisting the medics so you would have to mess up pretty egregiously to be sued.)
Like I said, I'm in New York State, here I can only practice under my prehospital certification (EMT). If you are thinking of working as a medic or EMT I strongly suggesting calling the sate department of EMS where you are going to be working and talking to them.