As the others have stated, PM docs come from all different specialities of medicine. Some states do have regulations on who can be a "Pain Management" doctor. My doctor is certified as a pain specialist. He's an Anesthesiologist. But many times it is a Physiatrist (Pain medicine and rehabilitation). Most PM's are MD's or DO's.
As someone else pointed out, a good clinic will offer different specialities. Not just an Rx clinic. Pain medicine is only one of many ways to treat pain.
You many times will need a referral to go to a PM, and once there that doc can refer you to other specialities if needed, like surgeons, PT, etc.
I would say the majority are Anesthesiologist's as dealing with pain is there speciality. But I have also seen just regular MD's dealing in PM. When looking for one, try to find a place that can offer many different types of approaches. There are places that just Rx, but that is limiting you to treating your pain or finding a cure. There are other PM's that will not Rx and do just injections (facet, epidural, etc). So in my opinion, the best place is one that offers all kinds of treatment.