If this were to happen, the person would become schizophrenic. This is because of the two individual personalities that occupy the same mind. It would be extremely difficult for someone to remain well and grounded. This sort of thing rarely happens. What is possible is that the aspect of the soul splits at birth and both are born at the same time in other parts of the world and may even have similiar looks or personalities. This does not happen as an adult. During the sleep state the other aspects will check in with it's counter part in the form of dream experiences and let it know what it has been doing as well as problem solve.
The other possibilty is a walk-in that would share the body but that walk-in doesn't stay unless the swap is permanent, meaning the original occupant of the body has left. This happens in near death incidents, traumas and even by choice of the soul.
There may be souls on parallel paths, which would give them common experiences and appear as if they are shared,but they are individual occupants.
If you look deeper into this subject, we are all aspects from the same source.