I have a 7870 2GB with 2 monitors and didn't notice any difference moving up from 1 monitor in games. I have used more than 1 GB of VRAM before so I would suggest getting a 2GB card. Which one depends on your main monitor resolution and games you play.
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.
Just be aware that you will take a small performance hit when playing games on a dual monitor setup. Even though when only one monitor is being used for games, the graphics card must also render what appears on the 2nd monitor. The performance hit depends on the power of the graphics card. The more powerful the graphics card the less the performance hit will be.
Based on a couple of benchmarks with 1920x1200 dual monitor setup the Radeon HD 5850 takes around a 5% - 8% performance hit simply when the 2nd monitor is connected.
A cheap dedicated graphics card for the 2nd monitor will eliminate the drop in performance.