Images using our image poster are all still here. Some off-site hosts require logging into our board to see the pictures, for some reason.
On Apache, the web browser is assigned to a particular user, often apache. When you browse as that user, you have permissions that world (anonymous) does not have. This works great when on the host itself, but when viewing from another host, not so much. When you log in, you change users on the server side, in addition to other viewing permissions. I suspect that your user on this site changes to apache when you log in, and the remote hosts run as apache, or something similar. Also, PHP has an option (as does your browser) to allow or deny cross-domain processes. In the apache virtual host, options can be set to control this behavior on a directory level, or via user, or by IP. If the remote host is running IIS, then yeah, results could be interesting going to apache, depending on how the IIS site was configured. Then too, there are web servers other than apache and IIS, I'm not familiar enough with most of them to comment.
On this site, the link does not show if it's determined to be blank. If the quest was able to edit, the img tag wouldn't show. It'd be like creating an img open and close tag with no content, saving, then editing. The img tag would be gone, only the guest can't edit. The img tag is blank, because there are no permissions.
I'm just guessing, here. I know nothing of how this site is set up, and only suspect a LAMP stack. This is just similar to something I dealt with previously.