Thad: Using just one monitor can be done, but it would be a major hassle just making sure you didn't accidentally show your opponent your view of the game. You might even be able to use the same browser, but I'm thinking probably not with cookies and passwords and all.

I used two separate computers logged into two accounts on It's Your Turn (IYT) about seven years ago to try it. Unfortunately my opponent didn't like Dark Chess and I had no one else to try it with. This game is on IYT as Walter_Montego versus Walter Montego. I had accidentally created the account when I didn't notice the underline was missing and somehow ended up with two accounts. It is the only game I ever used the second account for.

If you don't think it would be a hassle getting up from the monitor after you make your move, making sure to blank the screen, and then have your opponent take over for his turn, and them keep switching back and forth the whole game, I am not sure what you would think is tedious. It'd be a lot easier to use two monitors hooked to one computer. Your idea will work. Give it a try and see how well it goes.