You can usually sleep comfortably on your side or stomach; it depends a lot on how tightly the bivy is cut. For example, I found the REI Minimalist to be snugger than the Integral Designs Salathe. While I could sleep on my side in the Minimalist, the Salathe was a little looser and easier in which to do that.
I've never had any problems with condensation sleeping on my side in the bivy. However, I've never slept in it that way when it was fully closed: there's always been a fair amount of ventilation (especially with the Salathe.) Even in the rain, I've never closed it up completely since I use a small tarp, too.