well there may be a bit of ATI bias talking here but:
I heard a rumour the full cayman will have 2400 shaders. Ignoring clock speeds/die improvements thats a 50% increase in performance over a 5870. The 580 has 512 shaders, or 15% more than the 480 (which in itself imo was about 10-15% faster than the 5870, so for the sake of my rubbish maths we'll call it 30-35% over the 5870). SO that leaves ~15% performance to make up with clockspeeds and die improvements. I suspect the 580 will be more improved per shader over the 480 than the 6970/5870, but i cant see it being 20%+ more efficient (once caymans shader improvements are factored in).
Power consumption/Heat im expecting to be roughly the same...
So if i haven't made myself clear, my moneys on ATI for this round (sorry if my reasoning sounds like a load of BS. It purely speculation from me atm)