Can a user see the Active Directory computer description in the Windows network browser?
This is something that is only in AD, it does not get "pushed" to the member computers. So the end-users won't see it unless they open ADUC or some other AD query tool, as you point out.
It looks like you want an easy way to find which user is logged into what machine so that when they call into helpdesk, you don't have to ask for their computer name.
One easy solution is to create a login script (pushed out by GPO) which contains the following:
ifconfig /all >> \\server\share$\%username%.txt
Share$ needs to have write and modify permissions for everyone as the script is run as the logged in account. The script dumps computer name, IP, and every interface into a nice text file. You can go to the folder, search for the login name, and get the info you want.
A few approaches that come to my mind:
Query your domain controllers for connections from $user, and look at the source IP field for those connections (should be in the detailed view of one of the events for the connection or authentication).
Deployment of a tool like BGinfo that will display the relevant information in a place the user can easily locate.
Running a script to copy the description in Active Directory to the local description.
Just asking the user. It's not an elegant solution, but it's easy, and a lot of places still do it for that very reason - it's easier (and cheaper) to ask the user than to implement a system to eliminate the need to ask.