The JFBConnect setting for "Automatically Log Users In" is enabled on your site. I can see that by looking at the HTML on the page and seeing the following line of Javascript:
If you've disabled the Auto Login setting in JFBConnect and that Javascript still shows, it's an even stronger indicator that browser caching is enabled on your site and causing issues.
As for subscribing to posts, you can respond and click the "Subscribe" checkbox before you hit Submit. That should send an email after each post is made.
I hope that helps,
Alex