You working in the sculpt room? If so there is a Symm Copy option under the geometry menu. To use it turn symmetry on, and then turn on whatever planes you want it to copy to and then hit the button. It tends to copy the object side you last worked on, so if it copies the wrong way just hit undo alter a bit of the object on the side you want copied, hit undo again and then use it.
Edit: Sorry just noticed the other part of your question. If you want to save that part on the other side use the split tool and select that part into a new object, and then use symm copy and then merge that part back.