I install the Secure Shell client and server on all my Linux boxen (open Synaptic and install the "ssh" metapackage or, in a terminal as root, type: aptitude install ssh).
After you've installed ssh on both systems (I don't use Ubuntu; I use Debian... For all I know Ubuntu comes with ssh client and server pre-installed...) you can use Nautlius and choose Go -> Location and enter
Code:
sftp://[LAN IP OR HOSTNAME OF COMPUTER YOU'RE TRYING TO CONNECT TO]/home/[NAME OF THE USER ON THAT SYSTEM]
Once you're connected, choose Bookmarks -> Add Bookmark and you'll have the target system in the sidebar of Nautilus there for you to connect to any time you want!
The "LAN IP" will most likely be 192.168.x.x (let's use 192.168.1.101 as an example); so if the user's name on the other system is "jim", you could enter
Code:
sftp://192.168.1.101/home/jim
Alternatively, if the hostname of the system is "jims_computer" you could also input
Code:
sftp://jims_computer/home/jim
(I prefer to use hostnames instead of IP addresses as some routers will occasionally re-assign different IP's to computers on reboots or when swapping cables, etc.)
After you're connected and you have bookmarked the system into your Nautilus sidebar (so that we don't have to go through all of this again!) you can click on the bookmark in Nautilus and browse, drag and drop and even open and edit documents right from the other system as though the remote system were just another folder on your own system.
Being behind a wired router you don't have much to worry about in the way of eavesdropping, but traffic over ssh is all encrypted just in case.
The subject of your thread is called "Remote Desktop", and if all you want to do is control the other user's screen (you cannot transfer files this way; only literally control their mouse and keyboard as if you were sitting there) then you need to enable Remote Desktop under Preferences and use the remote desktop client (What does Ubuntu use now? Remmina? Vinaigre?) to connect to the remote machine.