There's actually some debate as to whether or not TRUE mpd exists.

It could be that the disorder is merely formed by those who semi-intentionally choose to forget who they are as more a defense mechanism to avoid stress/grief, etc. You know how if you are really stressed out you have sort of a different personality then when you are chill? The theory is that this is just a REALLY intense version of that. Also think of when you go into the kitchen but forget what you were going in there for. It was obviously important enough for you to get up and go there, but once you got in your brain switched to something else and blanked out what you were doing. Similar is the theory for MPD, which explains why people can forget years of events that were prior to them having the disorder, and why they usually "shift" during stressful points in their lives and the disorder usually occurs after a traumatic event.

As with most anything you do on a daily and regular basis I imagine this would also carry over into dreaming, a person switching personalities. However, hypnosis, a type of dreaming, is often used to force the personalilities to confront each other, which I imagine could also occur in a dream, as well as one personality reliving the memories of other personalilities, but I would suspeect that the person would usually be unaware of other personalities in the dreams just like in real life, and perhaps more so because of the lack of skepticism and general gullability of the dreamer.

Anyway, that's my two cents.