I assume you mean desktop icons?
In XP each user has their own desktop located in
"C:\Documents and Settings\<username>\desktop"
and there is also a shared desktop located in
"C:\Documents and Settings\All Users\desktop"
If the icons are in the all users desktop folder then they appear on all users desktops and if one user deletes them they are removed from all users desktops.
To solve the "problem" browse to C:\Documents and Settings\All Users\desktop select the icons you want on your desktop only and then paste them into your own users desktop folder. any which you want on your partners desktop put them in their desktop folder.
Or just delete them from either user account and then copy them back onto the desktop from the start menu while logged in as the account you want them to appear on the desktop.