Assuming we could do a human brain trransplant and it wouldn't damage too much the recipient body or the donated brain:
The body that is currently called "your body", when it woke up, would wake up with Stephen Hawking's (or whoever's) thoughts and memories, though he may be in different "moods" than he normally is in because now he would also have your endocrine system (your glands), which would affect his brain function.
That's because it's the brain that stores style of thinking, skills, memories and other types of mental content.