The other "self" you experience in your head during your dreams is probably a creation of your unconscious mind. When you dream, it gives you a kind of window to your unconscious which you wouldn't normally be privy to seeing through during waking hours.

I think the unconscious expresses itself through the right side of the brain because that's the side it resonates with. The things that we've pushed down into that rather inaccessible part of our consciousness are necessarily image oriented because they carry, or did at one time, enough of an emotional threat that we chose to bury them. Of course there's much more there in our unconscious minds than fear, but it still all seems to be stored in an impressionistic way.

I think in dreams we work on reclaiming this territory inside ourselves, confronting our demons as they appear (or running from them, in which case the same nightmares continue to haunt us) and in this way reducing their power over us.

In waking life they also have power over us - we just don't experience it directly as in dreams. Usually people can think of reasons for their actions and perceptions, but these explanations only cut the surface of the real why. There is so much more to ourselves than we can ever be aware of, but it seems like one of the best ways to keep finding out more is through paying attention to our dreams.