Given the scenerio of saving a file to their desktop, the file is on the local machine and can be found under the users desktop folder under their name under Settings and Documents.
Sorry to say but you really don't understand active directory or the client/server model.
A user does not logon to a "directory". They logon to a domain. By their membership and rights assignments in the domain, Active Directory sets what they can access and at what level.
I would suggest the following topics to google
client server model
peer to peer networking
purpose of drive mappings
These topics should give you a basic understanding of microsoft networking and how files/folders can be redirected vs just saving locally or via a mapped drive.