Don't get X Server and X client mixed up.
X Server is the bit that handles the display and input devices. It talks to the hardware to render the display. that's the X server. 
A X client is a program that runs in X. Every peice that makes up your desktop is a X client of one form or another. Mozilla would be a X client, for example.
So you could.. run a X Server on you local hardware. And then use SSh to connect using it's X forwarding features to run a card game on your school's computer and have the display output run to your local computer's X server. You could run a xterm on your school's computer while have the display output on your local computer at home, is another example.
So sure you can, you just open up the applications from your school on your desktop in either monitor. It'll be like your running it locally, you can copy and paste back and forth between programs, they will even use your local theme if your using something like Gnome and are using a GTK-based app or are using KDE QT-based app in kde. Except for the speed difference it would be like they originated from your computer.
In fact you can run dozens and dozens of apps from all sorts of different unix boxen on your home computer and it will be like it was originated from your home computer.
Now if you want to run a entire desktop from a seperate computer, on your local computer, along side, but seperate (like vnc) you can use: vnc, or use something like Xnest and just make it full screen.
Xnest is a Xclient that is also a X server. I suppose you can call it a virtual XServer. So it runs on your desktop in a window. But you can use it as a X server and run X clients WITHIN your Xnest'd server. Hell, it it'll even run it's own xscreensaver.
So in summary Xnest is a virtual X server that is actually a Xclient that can be run remotely over network OR locally that you can use to run X clients within Xnest's own X client on your local X server. It's enough to make you go batty. 
 check out this
check out this article for clarification.
Just remember: XServer is what handles the rendering and input and output from your pointer device, keyboard, and monitor. X clients are what runs as programs inside your XServer. 
XServer runs locally, but Xclients can be run from any authenticated source, on any computer that can handle the type of program (ie unix-like) and has a decent network connection to your computer. 
