Despite the common connection between the phrase "professional writer" and the image of the bespectacled social misanthrope grinding away at a typewriter to produce a novel or journalistic text, to analyze what sort of careers require and/or make use of professional writing as a key aspect of the field is much like analyzing what sorts of mammals make use of oxygen in their biological functions; an absence is infinitely more noticeable than not.
Let us, then, leave for now the discourse concerning journalistic and aesthetically literal professions as one well enough covered, and dive instead into professions that, on the surface, seem to have little to do with profession writing, yet rely upon it heavily. A prime (if obvious) example of this would be any position relating to the managerial side of business enterprises. This field carries with it canalized imagery of its own-- the puffed three piece suit with head sticking out of its collar, conducting dealings in brandy bars one moment and analyzing the hard, cold bottom line of a spreadsheet the next-- which tends to focus more on appearances, number$, and curly lipped rhetoric than on any sort of endeavor of endeavor of the written word. However, while there will always remain a certain truth in the assumption that most businessmen make more use of bluster than they do concise writing, this is to assume that their correspondence is limited only to others in the business world. To have gotten their initial enterprise off the ground, for example, it is a certainty that a written proposal and outline of that establishment was submitted to a bank or private investor. Moving on, as a business grows and money tends to flow in more directions, it is unavoidable that, even after a handshake and a spoken proposal to other potential partners and affiliates, more written proposals will need to be submitted to the variety of potential clients or colleagues one must woo in order to expand that business.
This essential principle-- that any effort to take or expand one's work to a wider audience/clientèle will inevitably require cooperation and, therefore, much written correspondence-- can be applied to absolutely any field or profession to flush out the role that professional writing plays within them. Mathematician? If he wants his new formula published or that fellowship at the best institution, he had better know how to write about his theories, policies, and resume' in order to convince others that he's on to something. Musician? Be it his organizing a tour, setting up relations with representation, or simply pitching his services to local venues, he's going to need to use professionally written materials to convince others that he's the next Bob Dylan and worth the time and effort. Baker? If that baker plans to shill his confections to more patrons than just family and friends, he's going to need to expand distribution of his goods to private functions or private businesses, which, of course, means writing rather persuasively to convince the newlyweds that his wedding cake is worth that extra C-note.
If skirting over the role of professional writing in industries such as journalism, education, publishing, or entertainment has left me with an incomplete assignment, I apologize. However, if some simply applies the same principle to these fields, the answers are quite self evident.