With my two-video-card setup I'm currently forced to run two non-Xinerama desktops side-by-side, each with their own gnome-panels. (When I move my mouse off the edge of one, input focus changes to the next one.) Additionally, sometimes I remote login to a desktop computer and I have two desktops running: one on the console and one connected to the remote terminal.
In both cases, I have not found a web browser that execute on both desktops at the same time. You may have exactly one instance per user per profile, and that instance runs on exactly one desktop -- not two or more. The only workaround is to run different browser profiles on each desktop (i.e. "firefox -profile $path-of-second-profile") and these separate profiles of course never share any data with each other.
I find this multiple-instance restriction with Opera, Firefox, Chrome, Midori, and Konquerer and at this point I've given up. In contrast, tools like Gnome-panel, Gnome-terminal, Nautilus, Thunar, Gedit, and Okular are happy to run multiple instances on different desktops simultaneously.