The permission rights on the Forum functions as for page templates, file- and image archive. That is if you have the right Write on the site you will inherit the rights to all forum on the web site. The folder"Forums" (and all underlying) inherits rights from the web site. In order to have rights you need to have the right to Write on the site alternative on the forum.
If you do not have the rights on the forum there will be a key on the forum/bubbles.
Since the forum inherits rights from the website it will also inherit the setting public/non public. If you have web site that is not public and you would like the visitors to read the threads in the forum the page the forum is located on needs to be public and the forum needs to be public. You can do this by right clicking on the forum and choosing Properties.