Thanks for pointing me to that issue, @IanM!
Okay, this is definitely a lead – it does have something to do with tags. Because when I temporary deactivate the "Tags" extension, the posts suddenly appear!
What's weird is that I don't see any wrongly set permissions, neither on the "Permissions" page, not on the "Tags" settings page.
When tags are disabled, I can also write a post without an exception. As soon as I re-activate tags, though, I'm back to the previous status (no discussions at all); I don't even see that one un-tagged post.