There are now multiple people who are currently surviving successfully with just half a brain. This has me thinking, about how the brain reassigns things depending upon what's available, in this case, this is a person with a personality, memories, thoughts, etc. in only half a brain. So where does all of this come from? The brain? The brainstem?

What would happen if someone were to remove half of their brain successfully and transplant it into someone else? Would that be the same person? Would it be a copy? Would the other bodies brain stem take over?

This is of course, if brain death didn't occur and the brain managed to stay functional for a while in the new body.