During sleep both hemispheres show large scale inactivity. There's no different in activity between the two hemispheres. All of our brain needs sleep. Just because only one half of our brain is responsible for speech doesn't mean we don't use it when we're awake.

It is my understanding that we have two different future-predicting brains: the limbic system (mammalian brain) and the neocortex (primates and dolphins brain). The r-complex is a purely reactive brain, and tries to predict little.

The neocortex is incredibly powerful, but the limbic system notices stuff that the neocortex misses on our day-to day routine. Things like micro-gestures (emotions drawn by our faces for mere fractions of a second), non-spoken language, etc.

This neocortex-unperceived stuff is perceived by us as 'feelings', the operating material of the limbic system.

The REM dream stage is the limbic system putting all those feelings as requests to the neocortex, in the way of: "give me a logical and understandable explanation for this feeling I've been having all week".

Then we experience crazy stuff that actually matches the limbic sensation, or we get the right answer. This also helps/interacts with learning processes.

For the rest of the dream state, I have no current theory/explanation.

We have only one brain.

what if you sever your corpus callosum?

Then we have two brains.

Two half brains. Everything has two halves.

http://en.wikipedia.org/wiki/Split-brain

That is a common misconception. In reality, there are other interhemispheric connections.