I have two PC's connected by ethernet, and linked via a Windows XP
"Network Place" connection.  What happens when I delete items off one
hard drive using the other computer?  For instance, I will be working
at one computer, open up a "Network Place" connection, delete a file
in that folder --- but then the deleted item appears in neither trash
can of computer one or computer two --- yet it is deleted.  Can
someone explain?

Files deleted over a network connection are not put in the recycle bin
(or trash can as you said). They are deleted from the hard drive
immediately.

http://www.microsoft.com/windowsxp/home/using/productdoc/en/default.asp?url=/windowsxp/home/using/productdoc/en/to_manage_recycle_bin.asp

(at the bottom):
The following items are not stored in the Recycle Bin and cannot be restored: 
Items deleted from network locations.

Careful, there. danmb3 is correct. NOTHING is sent to the recycle bin
EXCEPT files deleted from the hard drive. ALSO...if you have more than
one hard drive, each one will have its own recycle bin. So, don't
delete things off network drives, floppies, zip or other removable
drives without using extreme caution.