If you want to future-proof and are open to eventually running CFX then the 2GB would be the best choice... I'm running two 6870's (1GB) in CFX now and just started using Eyefinity this week (5670x1080 plus a bit of bezel compensation) and am pleasantly surprised with my performance even though I'm only running with 1GB of effective VRAM.
If you want to run full textures I'd say go for the 2GB card. Game's now don't tend to need more than 1, but BF3 and newer will need more at super high resolutions (I had to go down from ultra at 1920x1080 to medium to keep smooth rates).
I also would keep in mind the outputs on the cards you look at: both of those have 2 dvi and an HDMI, so you probably wont need any of those pesky display port adapters, although the $30 Accell one I had to get seems to be working just fine so far.
Awesome, yea I had messed with it a bit running two monitors... but it's pretty useless with a bezel in the middle so i picked up a 3rd last week. I wasn't really anticipating eyefinity when I bought my cards or I probably would have moved up to 2gb cards.