Yeah, in fact even if my whole brain were transplanted into another body, some of my mind would still be left in the old one.
I once had some surgery done in my abdominal region, and I had a spinal tap. This is where they inject novocaine into the base of the spine, shutting off all communication to the legs & other parts down there.
Afterwards, lying there in the recovery room, getting bored out of my wits, I tried desperately to move my feet. I couldn't yet, 'cuz the novocaine hadn't worn off. But it didn't feel like I couldn't move my feet - it felt like I never really had intended to move my feet in the first place!
The lack of feedback from the leg nerves was translated in my mind as "I must not have been serious in the first place". It took a lot of effort to overcome that feeling.
So, on that basis alone I think that transferring (physically or electrically or whatever) just the information in the brain wouldn't recreate the whole "mind". But then, if I was dying, I guess it would have to do.
Then you'd have two minds on the issue.