In common parlance, a 'professional' photographer means someone who earns their living through making images at the behest of a client (whether this is directly or indirectly - as in the case of stock photographers). The key to being successful at this is (a) to consistently deliver the results that meet your clients' expectations and (b) knowing how to keep your clients happy (there is always someone just as able who is willing to undercut your prices!!).
My experience of over 30 years of commercial photography (always with the aim of supporting my own personal image making) is that 90% is keeping your customers sweet (oh god - all of those interminable late evenings in restaurants being pleasant to people I would not otherwise have spent any time with) and, a very very important 10% ALWAYS delivering the goods requested.
I learnt the hard way on my own (with supposedly loyal clients who suddenly left despite the fact that they were always very happy with my work until one assignment was 'deemed' to be 'substandard' despite the fact that I had delivered outstanding results to vague briefs over many many years) but, observing friends who were in the higher echelons and being able to observe how their assistants fared, being someone's assistant was the best route to sustained success.
My personal observation is that, apart from core technique which, if taught correctly, can be learned in a few weeks, the area where an art/photography school can really excel is in either awaking you to the work of exemplary photographers' work that you were previously unaware of) or teaching you to formulate the nom-culture (in my personal view I would define this as bullshit) necessary to building a career where your 'clients' are galleries, funders and the wider art world.
Maybe I am just a dinosaur cynic but, having pursued my own work for the past 10 years and having had moderate success in gaining exhibitions and private collectors, I personally would like to return to the time when 'professional' meant more about your approach to your work and your knowledge of the subject rather than whether it defines how you earn the money to pay the bills.
So, after these rather unfocused ramblings, no, you do not need to go to school to be a photographer who wants to earn his/her living through photography but it might well help you in becoming a better (personal photography focused) photographer if your interest is in PHOTOGRAPHY as a medium rather than as an income stream.
Bests,
David
www.dsallen.de