The Vic is a training hospital and I think it's the case that they get students and graduates from three Scottish med faculties. Fife's a small place, so you're never really that far from home if you work there.
Any time that I'm at the Vic I see numerous trainee nurses wearing uniforms giving their designation as students from Dundee. (Can't remember how trainee doctors are identified - maybe via ID badges.) Once they're newly qualified, I think the docs just have the same ID badges as the rest of the regular staff.
I've tried Googling, but can only find specific reference to Lothian NHS Foundation Doctors being placed in Fife.