Hi Phil,
The trouble is....
Only Local Groups can contain foreign security principals (users from a trusted domain in this case).
Local Groups cannot contain other Local Groups. That means a local group on a PC cannot contain a domain local group.
By default, the only group with Administrative Rights on a domain member is "Domain Admins". Which is a Global Group (and cannot contain a user in a trusted domain), nor can it contain a local group on the domain.
Because of that lot, there's no reason you should have admin rights on a PC within a trusted domain unless you put something in place to take care of it. Hence the username / password prompt.
Chris