Hinesd :
Playing games on one monitor, browsing web and other smaller apps on other.
Assuming your monitors are 1920x1080, 1gb will work well, though 2gb does help. If you're just web browsing on your second monitor, 1gb is fine. If you'd like to say....watch a movie in HD on the other monitor while gaming on your primary, I'd recommend the 2gb card.
P.S. if you think you might want a third monitor at some point to game across all 3, you'll definitely want the 2gb version.