Joomla Search yields not authorised links
The problem
A link to a public article returns "You are not authorised to view this resource. You need to login." This problem goes back to Mambo days, and it still being fought with in new Joomla installations.
The issue comes down to the Itemid getting lost. The itemid for an article lets Joomla know where in the menu tree this article is being pointed to, and what the bread crumb path should look like, if you're displaying it. If this item is missing, it should get set to the default itemid for your home page. Sometimes it gets set to null, or 0. Then you'll start seeing the 'not authorised' message.
This may happen if you have static content that isn't linked from any of your menus. A work around for this problem is to turn off 'Static Content Non-Menu' in your search content mambot/plugin, then create a non-published menu and add all static content items to this menu.