The desktop.ini files store information regarding the layout of the desktop or display options of a folder for example to arrange the icons in alaphabetical order.

The reason why you normally can't see them is because Windows Explorer is hiding them from you but you can view another user's because their's aren't being used by Explorer. If you try to delete them Windows will replace them as it saves the default folder/desktop view options.
You can also force Windows to show them for you from a folder windows choose Tools>Folder Options>File Types and uncheck "Hide protected operating system files"

Really for security reasons (read to help protect you from viruses) all users should use limited accounts (this is even more important if they have Internet access), this also means that they can't see any other user's files unless you're the administrator, the only disadvantage is it breaks some old software - the choice is yours security vs backwards compatibility.

[bitch]This is one of the annoying things about Windows
[.ShellClassInfo]
LocalizedResourceName=@%SystemRoot%\system32\shell32.dll,-21787

wtf is -21787?[/bitch]