Comment 15 for bug 844081

Keep in mind that LightDM should probably fall back gracefully if it
can't find the background image for a user (and if it doesn't that's
almost definitely a bug). So, if you're on a multi-user machine and
you're concerned about other people seeing your stuff, the best
solution (and the one we support) is to change the accessibility of
the file itself.

So, Nathanel, the setting you want is in Nautilus. Right click the
offending file, choose Properties, then head to Permissions and tell
it to disable access for Others.