Intrepid web developers reading this comment, please note that CORS is not, in fact, a protection mechanism. It's a way to relax the Same Origin Policy which is actually the protection relevant here. You don't need a CORS policy to protect a site from cross-site attacks, you need no CORS policy. Go ahead and make your little chat app, you're not at risk of having your messages stolen because of a lack of CORS headers.
I did say it wrong, but my point was that the site doesn't segment off each "site" into a different subdomain or any other ruleset that would allow the same origin policy to restrict access.
As it is with this site, the messages can get "stolen" by any other site on the same domain, which can be anything since anyone can upload one and direct a victim to them.