In order to answer this question you need to define the terms "profession" and "organisation". I would submit that while I have met many people in EMS whom act in a professional manner, we are a vocation and not yet a profession. Our barriers to entry at both the paramedic and EMT levels are quite low. We have little control over the direction of changes in clinical care, and ultimately report to EM/Cardiology.
There may be countries where the move towards professionalisation is more advanced, e.g. some parts of Aus/NZ where Bachelor's degree entry-to-practice is becoming the norm, or the UK with their consultant paramedic and paramedic practitoner roles, parts of Scandinavia where the paramedics are essentially nurse-practitioners, or the countries where EMS is provided at the advanced level by specialist physicians. I doubt anyone would argue that an anesthetist working on a French SAMU ambulance meets the traditional definition of a professional, for instance.
We talk a good game in EMS about how professional we are, but when you can become an EMT in less than six months, and most paramedics have less education than a plumber, it's not that meaningful.
This isn't a slam on the vocation/occupation, more a recognition of the opportunity for growth and transformation.
** Or a slam on plumbers, who intermittently stop my house from flooding. But if all we define professional as, is someone who makes money from a specific job, then "professional paramedic" may mean little more than "professional exotic dancer".