Conversation permalinks are running snapshots of an existing conversation that may be shared and viewed only through a secure URL. This is useful when you need to share the contents of a thread without giving access to a party reply-to privileges. You can modify the privacy setting for your permalinks by going to "Brands" then "See all settings" and finally "Permalink privacy settings".
The following page will allow you to hide the display names of the parties involved as well as attachments if there are any. Third parties viewing via the permalink will not have access to this information.