When you forward X you connect with -X. It should connect to your home directory, the one on the remote machine. Try pwd once you have connected.
There you can run programs on the remote machine, but the display will be on your local machine. So you could run Nautilus on the remote machine and manipulate files there. But you cannot move files between the remote machine and the local machine that way.
ssh -X firstname.lastname@example.org
No graphical programs on the remote machine access your local machine, but there two easy ways to transfer files. To transfer files, you could run FileZilla, but you could also use Nautilus. If you start Nautilus locally on your machine, press ctrl-L and then enter the URI for the remote server: sftp://email@example.com/