Okay, well I'll prefix this with the disclaimer: I don't use and never have used NFS. But my understanding is this: NFS allows mappings between users on the client system and users on the server system to determine what access (if any) the client has. If on your Unixx system, files are owned by "Joe" and NFS maps this Joe to Joe on the client system, Andy, say will not have access.
That's simplified because I've not considered group ownership, but you get the point. Linux file systems have three sets of access permissions: the owner of the file, the group owner and everyone else. Check the file permissions on your Unix box. If you see something like "rwxr-x---" generally speaking anonymous access will not be enabled
unless the anonymous (group) user for NFS is an owner of the file.
I don't think I've explained that very well, but I've found a more detailed explanation here:
http://www.troubleshooters.com/linux/nfs.htm. It relates to Linux, but the same principles will apply to your Unix system.