In such a context, your species sounds more like two individuals that share a body in alternating times than one individual with dual personalities.
Of course, such distinction is arbitrarily defined by the chosen biology and planned society:
At one extreme, you will have two people that lead completely different lives in different times and may interact very little, like a flatmate or, in your case, a "bodymate".
Society-wise, law enforcement may play a role to prevent hostile interactions between the personalities, which would be seen the same way as attacking someone in their sleep.
Conversely, others may choose to share a lot and interact closely, like a sibling.
With that said, your gamut of relationships may be as wide as the one for two people in our reality.
However, of course, there is one huge caveat: they do share a body and, as opposed to flats, changing it may not be an option.
As such, I think most relationships will tend to be collaboratively and harmonious.
After all, if you assume that your species was subject to
some kind of evolution through natural selection, rivalling bodymates would probably not benefit from self competition and such characteristic would disappear.
If your species is human-like or have an analogous human-history, I think that would make for really interesting plots:
Surely, sharing bodies would have a huge impact in society and culture and would probably develop a whole set of social facts.
"Bodymates" would define a new kind of kinship, which may influence humanity more than siblings or parents do.
At the same time, it would also affect the other 'regular' kinship. After all, everybody would have two mothers, two fathers, etc.
Regarding communication, the time constraint would be pretty solid at pre-writing times.
Without distance (being it temporal or spatial) communication, both personalities would probably rely on biology tricks to communicate:
maybe the awaken personality could influence the sleeper's dreams, or hormones could be used to regulate mood, behavior and exchange information, in the same way that hormones work in our bloodstream to orchestrate body functions.
With the advent of writing, more complex and nuanced interactions could have developed:
Written communication would enable rich discussions and agreements, allowing 'bodymates' to become anything from acquaintances or business partners to close friends or platonic lovers who share an epistolary relationship.
With the onset of more advanced technologies, these relationships may indeed approach the level of generic relationships with other people. If the sleeping times of both personalities is roughly equal, written communication may somewhat be similar to what we have between parties in different time zones: huge communications delays between distant time zones is well understood and expected, and it doesn’t render communication impossible.