One good solution is to have a dedicated machine for the tests, which you can continually monitor and can easily be updated with changes from your dev machine. Since that isn't always an option, I'll present a few alternatives for several different platforms which allow you to run UI tests without having a separate machine to run them on.
VNC is a great tool for controlling remote computers of any platform. If you have fast user switching enabled, you can actually use OSXvnc to display the desktop for a user other than the one currently using the display. If you configure OSXvnc to not quit when switching to another user, you will always be able to connect to the desktop where OSXvnc was first launched.
Set up a "test" user, and set up OSXvnc to launch when that user logs in. Set up the "Startup" tab like this:
Once OSXvnc is running, you can access the "test" desktop while using your normal desktop. Run your tests in the background, peek in on them to check the results.
Run vncserver, either manually or as a daemon. This will give you an additional X display which is totally separate from your physical one. Connect to the new display with vncviewer, and you have a window onto your "test machine".
Windows isn't quite as functional when it comes to sharing a display, but you can still get some mileage out of UI testing on the same machine. There's a peculiar little mode in windows, I'll call it "service mode", where you think you have a display but you don't. This prevents java.awt.Robot from working properly, but you can still get some mileage from UI tests that don't depend on screen captures or native drag and drop. Abbot (a UI testing library) will work just fine in this mode, and does so by artificially generating events directly into the AWT event queue in order to drive a UI.
The easiest way to run tests in this manner is to install cygwin sshd. It runs as a service, so by default it doesn't have access to the desktop (there is a checkbox in the services admin control panel that indicates whether a service is allowed desktop access). Once that is installed and configured (google "cygwin sshd install"), you can use ssh to get a shell prompt within the service mode "sandbox". If you've got an ant script to run your tests, just invoke it. Abbot automatically detects whether this mode is in effect, so if you're using that library, everything works out of the box. If you only run your tests interactively through an IDE, you're out of luck, since you won't be able to interact with the IDE if you launch it in service mode.