Happens on empty cache and regardless what values I put in the form fields. I'm stumped.
(See also this thread.)
We had a similar issue and tried several tests to resolve the issue. Checked all Anonymous Roles to YES, not this problem. Turned off all WordPress Plugins, not this problem. Changes theme, BINGO!
There was a conflict with one of the JQuery/Lightbox type page overlay option within our theme that cause the forms not to submit. Removing that option fixed the issue.
The overlay window was loading a separate page and somehow it was loading the Registration form too!
So, Theme conflict in WordPress.