You have to leave the original X server running. You can start another X server on another tty. So, on a typical system, do ctrl+alt+f1, then log in and run
startx -- :1. You should end up with another X session on reachable by ctrl+alt+f8.
Any number of X servers can be started by changing the number after the colon; if you use a number larger than 12, you can use
chvt to change to it instead of the key combo.
If you want, you can setup special
.xinitrc files that start different desktop environments. So you might have a
.xinitrc-kde that starts a KDE session. In that file, you'd have something like
exec startkde. And you'd start X like by doing
startx ./.xinitrc-kde -- :1.
If you plan on running Firefox on both the sessions, there may be some issues. Look into the "no-remote" and "ProfileManager" command line options for Firefox.