I just installed the latest version of AEC (18.104.22.168) and it's not working. The links show up, but clicking the Edit link results in a blank overlay popping over the page. Using Firebug in Firefox to view the generated HTML shows that nothing is being generated (image) by the URL wp-content/plugins/wp-ajax-edit-comments/php/comment-editor.php?action=editcomment&p=19&c=832&KeepThis=true&
This is true both for admins and anonymous editors. (I haven't tested non-admin registered users.) I don't see any errors in my HTTP logs. I'm on WP 2.8.4.