Bert,
Mozy allows you to restore folders, you can't see the files inside of the folders.
Theoretically you can upload a file and let somebody else download it, but in such case you would have to give your username and password to another person, who would login into your accound and download the selected folder.
That person would have unlimited access to all files and folders in your backup, so it would probably be a HIPPA violation.
It is better to use Logmein for large files transfer.