How do I (properly) access another users' recycling bin without logging in as him?

1. Start an elevated copy of notepad -- or well, any program that has a modern Open dialog. (press the Windows key, type notepad, press Ctrl-Shift-Enter)
2. From File choose Open.
3. In the location bar at the top, click on the icon with the folder to switch to input mode and type the following (literally): Recycle Bin, followed by Enter.
4. Change the dropdown at the lower-right of the Open window from Text Documents (*.txt) to All Files
5. Lo and behod, the contents of $ADMIN's recycle bin!

Don't forget to close notepad when you're done!

(I agree, this is not really a 'proper' way, but I guess it's better than blindly wiping everything.)