Since modern PCs can use two screens at the same time I wonder if it is possible to plug in two keyboards and mouses as well to have the two screens run two (more or less) independent X-sessions at once?
In short, yes, this is possible. The relevant search string you are looking for is "Multi-seat X".
The Ubuntu wiki, Gentoo wiki, Debian wiki and Arch wiki all have articles related to multi-seat X. A number of other articles can be found on the Xorg wiki page on multiseat and even more can be found on google.
From what I can tell from these articles, there are two ways to do this:
Which of these methods will work for you will depend on the version of Xorg you are running and your hardware. Multiple X servers seem to be the easier route if your hardware setup supports it. There is also work to be done with the display manager, sound server, and other components -- much of which is covered in the various articles linked above.
Additionally, there is a multiseat display manager (MDM) to automate these configurations. It's not yet integrated in most distributions, though (the