Well, it definitely does happen, as we see from the Prologue of AFFC:
“Perhaps he would do better to remain on this side of the narrow sea. He could buy a donkey with the coin he’d saved, and he and Rosey could take turns riding it as they wandered Westeros. Ebrose might not think him worthy of the silver, but Pate knew how to set a bone and leech a fever. The smallfolk would be grateful for his help. If he could learn to cut hair and shave beards, he might even be a barber. That would be enough, he told himself, so long as I had Rosey. Rosey was all that he wanted in the world.”
However, I don’t think it’s that common, in part because the Citadel has a strong institutional incentive to maintain its monopolies. The Citadel does seem to have certain exceptions: acolytes who’ve gotten their link (and thus maintain the monopoly) are allowed to work as scribes for the general public:
“Just beyond stood Scribe’s Hearth, where Oldtowners came in search of acolytes to write their wills and read their letters. Half a dozen bored scribes sat in open stalls, waiting for some custom…” (AFFC)
But I think the Citadel would probably come down hard on anyone practicing more refined specialties than mere writing and reading; I would imagine medicine and ravenry would be particularly jealously guarded because they are some of the more important services that the nobility need. So Pate might be able to ply his trade as a barber-surgeon among the rural smallfolk, but he might well be hauled up before the lord of a significant holdfast or town for practicing without a license (if only because the lords don’t want to be blacklisted by the Citadel).