From what I understand, there's actually nothing wrong with this.
There has been research into what is known as "Split-Brain disorder", which has looked at patients who have undergone a hemispherectomy and ended up displaying some unusual behaviour.
It's worth noting that most hemispherectomy patients are actually fine post-surgery, with many regaining their full cognitive faculties shortly afterwards and returning to society with no long-lasting effects.
Having a split personality is a rare, but possible, outcome of having a split brain. The human function mainly affected by such surgery is actually language, so the fact that one side doesn't control speech, at least to begin with, is absolutely correct.
What you most likely end up with is the left half of the brain controlling the right half of the body, and vice versa. Over time, it is also possible for the two "identities" to become aware of the other and even to communicate, once Neuroplasticity has run its course and the two halves are on roughly equal footing as far as cognition is concerned.
It's quite likely that each half won't have full access to all their memories of the person they used to be before, and it's also possible that one side may even be completely emotionless to begin with, but as with everything else, with time they will each become what you or I would regard as a normal, fully functioning human, dual personality aside, due to plasticity.
In this particular case, there's a good chance that each side won't actually "feel" the other's presence any differently than they would "feel" the presence of any other person. There are plenty of people with multiple personalities (Dissociative Identity Disorder), and in some cases none of their personalities are aware of any of the others. When they switch, they may or may not retain memories of what they were doing beforehand, but if they do, they often believe it was that particular self that experienced them. They remember doing it, so who else could it be?
I can tell you, however, that if you are aware of an additional personality, it is entirely possible to have a conversation with them. You will often end up finishing each other's sentences, and you can tell what the other is feeling as they are "talking"(thinking) to you.
Ultimately, it varies enough from person to person that you can come up with just about any scenario, whether they know of each others' existence or not, whether they can converse or not, whether they can remember what the other experiences or not, etc. All these things are possible.
: I took a Neuroscience module at University, so I should know what I'm talking about. Probably.