Hope this helps
We have one 2003 server running iis6 with sharepoint services installed and three 2000 servers with. Each is a separate domain with external trusts linking between them all. Within each of the 2000 servers I have added a host record with the ip address and header name of the sharepoint site within dns in the primary zone. Each server also has secondary zones added for each of the other domains
I then added 3 crosssite groups, one for each of the 2000 server users and added all staff that needed access (this gets round the need for login screen.
I presume from your post that you are running sharepoint team services (the forerunner to team services). Sharepoint sevices is a 2003 server product,it requires 2003 server, sql server, and iis6 (check with microsoft website to be sure)
Sharepoint team services runs fine on 2000 server I have it ruuning on test at mo. Here are some thoughts for you I don't use NT much so hope info is okay:
1) if you are not running active directory you will probably need to set up a list of users.
2) Check dns, if there is a connection between servers ie you can ping by name and ip address then try creating a host header record for site. Then enter this as host record in each of the dns servers that can't see the site with the ip address of the site.
Hope this helps and does not lead you down wrong path. All my problems were cleared up simply by getting dns and trust relationships right.