Looks strange to me. I think that sitemap works fine and does not lists the listings you are not authorized to access.
Maybe you created the sitemap when you were logged in and now you see a cached version.
First check the access to the menus items that contains content for registered users. You most probably have a wrong setting there.
If this is not the case logout from elxis. Go to cache folder and delete the sitemap's cache falies.
Now visit the sitemap's URL in order for the sitemap to be re-generated: http://www.example.com/google.xml
(SEO PRO URL)
Check the sitemap, still lists content for authorized users?