I did a bit of digging and found that most sources agree that plague doctors were, "normally either second rate doctors who had not been particularly successful in their practice or young doctors trying to establish themselves." It appears that the in the majority of cases an agreement was reached between the town's administration and the doctor in question. The most notable case I could find was that of Dr.Ventura who was to act as the plague doctor for the town of Pavia. The majority of the contract deals with salary and the providing of a house to this Ventura. According to the book The Medieval City there was actually much dispute between Ventura and the town officals over clauses two and five again in regards to salary, so it appears Plague Doctors weren't too well treated despite the work they were doing.

Obviously the problem with the Plague was that it was highly contagious and anyone treating someone suffering from it ran the risk of being affected themselves. The solution to this was "physicians or surgeons, especially hired by an infected town or village in time of an epidemic" who "were responsible for the treatment of the plague patients only and had to refrain from intercourse with the rest of the population." Considering the risk of death and low apparent pay it's strange that any doctor would even consider this role but if we jump back to Dr. Ventura a moment it appears that - according again to The Medieval City, it is doubtful whether he assumed this risk for purely humanitarian reasons. Thus it appears that the majority of plague doctors weren't the most wholesome of fellows.

But I've gone off on a bit of a tangent there, so to bring it back to your question: Plague Doctors had to be trained Physicians, but not neccassarily very succesful ones. To become a Plague Doctor one had to reach an agreement with the town they wanted to work in, the town would then provide them with a salary and most likely some kind of accomadation if they weren't already residents. Considering the crisis that was happening the towns probably didn't care much for how reknowned the doctor was, just that they did have an education and were willing to put their life on the line without interacting with anyone put the infected.

Sources: I don't know how trustworthy this site is generally, but its information reflects what I found elsewhere and it sums up Dr. Ventura's contract nicely and The Medieval City (edited by Harry A. Miskimin David Herlihy and A. L. Udovitch) - Chapter 4, A Plague Doctor, by Carlo M. Cipolla