First you need to enable auditing on the server in question...
Start->Control Panels->Administrative Tools->Local Security Policy
Then, Navigate to Local Policies->Audit Policy.
In there, open "Audit Object Access" and select "Success."
Now, navigate to the root folder of the share, right click, and choose Properties. Click on the security tab, and then click "Advanced."
Change to the "Auditing" tab. Click "Add" and enter the username or groupname whose file access you suspect... "Everyone" will obviously grab everyone. Check off the "Success" checkbox for "Delete Subfolders and Files" and "Delete" option.
Then watch your Security log. Any time anyone deletes a file or folder successfully, it will be logged there.