I'm going to purchase a gigabyte AMD HD 7790. I'm going to use dual monitor at 1920x1080 for desktop applications (For games I will be using only 1 monitor).
1gb is probably fine as that GPU without xfire isn't strong enough to push graphics past that, could go for the 2 to be safe if the cost isn't a big deal. If you aren't gaming at 1080p then I would for sure go with the 1 gb.
For any modern gaming, even on a single monitor, 2GB of VRAM is ideal. Depends on the games you will be playing as well. 1GB might be fine for that card though.
The difference in price is pretty small, and in the long run (and even now, depending on your game and quality setting) 2gb of vram will definitely be a big help.
What I understood from all yours answers is that 2gb isn't a MUST have for dual monitors, it obviously helps. I also found out that in some games the 1gb version outperforms the 2gb version, it is weird.