Lets take your example to a hypothetical extreme
Just imagine if one half of the brain could be transplanted to another body - as with other organs.
Would you exist simultaneously as two separate people? - I think not.
Good question - and gets to the heart of what defines a person. The consensus seems to settle on higher neurological function, hence brain death being considered to define the death of the person even if other organs can be maintained.
But there is also the consideration of neural continuity - in other words a person is someone with a unique and continuing neurological function. So I'm not you can make such a glib 'I think' not answer.
Let's try a slightly more relevant sophisticated thought experiment - Imagine someone is brain dead (call this person A), but all other organs are functioning. You remove the dead brain and replace it with two half brains from two separate people (call them persons B and C). Let's assume there is no connectivity between each half of the brain as percorpus callosotomy, but that each half brain is now able to interact with the rest of the body so we have a living body that is no longer brain dead.
Who is the person or persons thus derived - it is person A - I wouldn't think so as their brain is dead and gone.
Why would we think this is person B rather than C, or C rather than B - surely both B and C are equally present. Or is it person D.