Same problem. I am using Google Chrome and WordPress 3.6. It sends the e-mails from the back-end if you use that, but the front end option doesn’t send the e-mails.
Dan,
Thanks for your feedback on this. It looks like it must either be the Theme (DirectoryPress) or some custom code that I have had written which directly affects the registration pages. Unfortunately, this is low down my packed priority list at the moment, so I’ll have to come back to it in due course, but thanks for your investigation. I’ll let you know if I find anything.