In a world I am building, the days last 9 years. One of the results of this is that many species have evolved to sleep like dolphins, one half of their brain at a time. The two sides and thus two personalities never are conscious at the same time. Everybody is like this.
If such a system were to exist, how would these each personality keep track of what the other is doing while they are unconscious?
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.
If the process is really like sleep, I assume the brain does not "switch personality" suddenly, unless it is very tired. In real life we do not simply shut down and start sleeping randomly. We need schedules and (most of us) some time laying down quietly before we get to sleep.
This schedule means that in most cases a situation in which Personality 1 "wakes up" and discovers Personality 2 has taken him somewhere he doesn't know simply won't happen. It is as likely to happen as it is to you to wake up in an unknown parking lot without any idea of how you got there.
Your problem now changes from "waking up somewhere weird, not knowing what happened and/or is happening and having no idea of what to do" to simply "waking up on your bed just like any day and wondering what your body did during the night". Of course the idea that our bodies are strolling around without us is terrifying for you and me, but your creatures should be used to that.
Unless one of the personalities does something which might in some way endanger the body, I don't really see a need for the personalities to talk. Of course they can, if they want to, but there should be no need.
Things get a little different when you start to think about interacting with other people. In most cases, you need to know whether you're talking to Personality 1 or 2. This will probably result in some sort of simple ritual before any real interaction starts. Simply saying your name before starting a conversation should be enough.
The fact that everybody is sharing brains makes for great story potential and would make for a culture very different from what we're used to, but I don't see communication being a problem.
Basically you're going to have to spend some time in catching up on what you already did today, whether that's in the form of reading a diary or a blog, maybe it's a one-on-one personality debrief, or someone else telling you about it. If you don't have that then you end up in a Memento situation, constantly waking up not knowing where you are or what's happening, that's actually workable if your two personalities live completely separate lives though. P1 wakes up eats, goes to work at the factory, comes home eats and goes to bed then P2 wakes up eats and goes to teach at the polytechnic comes home eats and goes to bed. P1 and P2 have similar skill sets, P2 teaches what P1 does for a day job, but completely separate lives lived on separate cycles that never overlap.
