My point is that it just feels like you have one mind: but in reality "you" are just a series of neural patterns that share a sense of continuity. Bits of your brain light up, other bits go dark, but "you" persist.
If we put bits of that "you" into two heads (or two separated hemispheres in the same head), then we shouldn't be surprised if both bits share a sense of continuity with the original "you".

Grant Hutchison