Hi Nik
I am not a sharepoint expert, however from what you are saying it sounds like a permissions problem. Without knowing the in's n out's of your network, I would assume that all the internet users acessing the sharepoint site have permission to access the site but not the required security privilages in AD!
Are you talking about annoymous users? If not, check these users security settings against some of the files an see what it says.
CP