Well for me, 2 of those answers point in 2 different directions.

It doesn't work when they log on to your pc - indicates profile issue, not hardware
It doesn't work when you log on to their pc - indicates hardware, not profile issue
Unless you and the other group connect to different servers or something, I have no more ideas. To explain, a network drive letter (volume label) could be different (e.g. F: for you, G: for them, or both the same (F) but different servers) but your database is configured to connect to tables using F on your server. Then it wouldn't work for anyone who's not mapped to the linked table paths regardless of where they log in.

I think that's all I've got, because it seems more like a network or folder permissions issue and those aren't my specialty.