oh boy! I know exactly what you mean!! my girls will play very happily together but as soon as they are in competition for a toy or for my attention - it's ON! biting, pinching, hair pulling, pushing. It doesn't matter if I'm right there on in another room.
We use time out here (they sleep in beds now so they have time out in a cot) which works but it is hard to be consistent - especially since I don't have one who stands their ground and one who runs to me and cries - they both stand their ground and try to get a mark on their sister until one of them is too hurt to continue or I intervene. So most of the time I don't really know WHO to send to time out - since I can't tell who started it or what caused the disagreement. Usually I just time out both of them!

Anyway, you're not alone! I'm sure it will get better when they learn to talk to each other instead of using (rather forceful!) body language! I have noticed that even though they are talking quite a bit now, they still don't really talk to each other much - they prefer to communicate in their own way. So I am hoping words start flowing between them sooner rather than later.