I'm also able to replicate this issue in my vB 3.8.5 but it's highly customized and have a lot of mods installed so I can't say if this is a vBulletin problem or a problem due to a mod.
Issue is that New Posts is displaying announcements in this way:
http://www.forosperu.net/announcement.php?f=XXX&a=YY
Where XXX is the forumid of a non public forum (Staff only) so that's why people are getting a no permissions error. I don't know why New Posts is displaying the announcement with the forumid of my Staff only forum.
I think that in a default vBulletin 3.8.5 installation New Posts don't display announcements. Anyone can confirm this please? Thanks.