"A lawyer who represents himself has a fool for a client". I think this extends into other professions as well, not the least of which would be tattooing. Minimally, the art would suffer due to physical restrictions - how many painters paint with their off-hand, one eye closed, gauze over the other and balanced on their toes?
Can, yeah, I'm sure they can. Should is another matter entirely, and any artist who'd work on themselves I'd be a little leery of going to to work on me.