1000w for a single 5970... overkill central but good to have for the headroom in case you get an 980X or something. I can understand two, but these cards aren't that power hungry.
As for the Seasonic s12 550w you'll be perfectly uber fine with just that PSU. Remember, these cards aren't power hungry at all.
The 5870 only uses about 190w max, so you should be good to go depending on what else you have hooked up. My system at full load(CPU and GPU) uses just under 500w, and I'm pushing a lot of voltage into my Phenom II.