OK, so really just "another X11 session". I think you could do this via Xephyr or (possibly) XNest, however particularly if the software gets the idea that it has to work over a network connection rather than using shared memory this could be slow... seriously slow, I thing it's something that crops up every so after where the widget set is polling the input devices excessively.
Have you tried tunneling over SSH? You'll probably have a single window rather than the full desktop, but it might do what you want and will be relatively painless.
VNC will generally perform better, and these days is fairly compatible in terms of supported APIs. If you're running lightdm as the display manager you'll find that it has the facility to map a root window (i.e. with the full desktop environment) between systems, otherwise while I don't know the "proper" way to set this up I've somewhere got scripts that will make a VNC session available on a predetermined port completely separate from anything else going on on the same system.
Finally, if you use Leave -> Switch User it will give you a completely separate session on <Ctrl><Alt><F8> with your original still on <Ctrl><Alt><F7>. However KDE has a quirk in that it won't actually allow the same user to login twice (i.e. you'll need to set up a second local user and designate a shared-accessible directory), and my experience it that you /really/ need to prevent either session from going into a screensave state since getting out can be tricky.
Quite a lot distilled into a handful of paras there...
MarkMLl