Are there any security risks by allowing another machine to use my computer as an X server?

Of course.

This can be minimized by using ssh to pass the X protocol. The advantage is that the data is encrypted. There is also a bit of compression going on...

ssh also sets up unprivileged keys so that the remote host will be unable to do certain things. One is that you can't remotely lock the screen, but there are other things that could happen.

Some of the issues that will remain is that audio devices don't work (that isn't part of the X protocol). And if you do manage to enable remote audio (it isn't exactly impossible - but it requires port forwarding which is usually disabled, and a client/server connection to forward the data. This would usually introduce too much delay).

Why would your machine be an X server? Is this some sort of Demo on your machine, that is run remotely? (verses running VNC or the like) Or do you mean running X client on your machine? (server is the one where X appears)

It is an X server if it displays windows from an application using X for display... And it doesn't matter whether that application is on the same machine or one across the country.

The disadvantage for the "across the country" is that the straight X connections are not encrypted - thus you use ssh to provide encryption, and it tunnels the data from "across the country".

NORMALLY, Linux X servers do not support TCP. This is because of the insecurity that was imposed from way back (encryption was a munitions). ssh provides a tunnel that provides the encryption - and passes the packets to the X server via the UNIX domain socket used by the X server.

Maybe I should refraise it. I am trying to figure out what GUI application, is going to be run on his machine, where the user/operator can't see the results.
How I get, the why, not so much.

None.

The GUI application runs on the remote machine, using the ssh connection to forward the X packet to the local X server running on his workstation.