I know it can run on 550 watt because I did it
I upgraded to 750w though, because I wanted a better PSU then the 550 watt that came in the case.
RE: those cards, well, the 6870 is better out of the box. Now if you overclock, the difference blurs. But both of them hit the same overclock limits, since they use the same chips. Only the stream count is different. I have the asus 6850, and it clocks well past the stock 6870. I outperformed it benchmark-wise every time, getting up to 960mhz core, and 1150mhz out of the memory. So to recap- if you OC, the 6850 (may be) better, and if you don't the 6870 is the winner. Price is king to me, so that made it an easy decision.
Also note reference design for the 6850 and 6870 is for 1x PCIe power connector, although almost all the boards I've seen use 2 PCI connectors.