It depends upon if you want the simple or extended definition of professional. The most basic definition is someone who gets paid for playing. But in some circles the term "professional" takes on a rather lofty definition that includes a range of characteristics and duties that members are expected to meet and fulfill.
I did a Google search for characteristics of a profession and this page was about the most succinct:
Characteristics of a Profession
In order to meet the loftiest definition of professional musician you would need a formal education and to belong to a union and other musically related professional organizations. You would be extremely competent. You would be ethically guided towards providing a quality service to clients as well as maintaining the health of the profession.
Consider a continuum of professionalism with laymen and unskilled laborers at one end of the continuum and highly trained/qualified professionals at the other. Anyone who get's paid to perform to play music is a professional in my book, and some musicians exhibit a higher degree of professionalism than others.