It depends on the school...the candidate....the internship and the residency.
I managed to obtain a small animal internship at the school I completed my DVM at....but it wasn't the easiest thing to do. And in fact they aren't taking "one of their own" for this coming intern class...
My situation kept me here (husband/med school) but if I had the opportunity to leave, I'd take it.
There's a lot you can learn by travelling to other schools. Even in externships along the way.
Some of the residents here were former students who went somewhere else for internship. That's more likely than staying for an internship. However, some private practices will keep their interns for residency spots.
Most of what I know about is small animal related. Equine and food animal tend to play by their own rules sometimes...so you might want to talk to other folks about those...
Competitive specialities are competitive all around so even if you think you are well liked and "have an in" you probably don't.