Can you explain a bit more about the problem? The SCLogin login area uses Joomla's standard login process. When you fill out a username/password on the form and click "Login", your submitting an HTML form. There really isn't much that the SCLogin module can be doing 'wrong' there.
On your page linked to above, I was able to click the login button with an empty username/password and the form seemed to submit just fine. If there's more to how I should be testing, please let me know:
* Do certain credentials fail?
* Is there an error message displayed?
* Is the home page the proper spot to test, or I should I try from a different URL?
Sorry for the delayed response. The issue is with something related to your SEF URLs or your .htaccess configuration. The login and logout forms are trying to POST to the current page, which is normal. However, there's a redirect occurring which is preventing the posted data from properly being processed by Joomla.
The SCLogin module simply uses Joomla's standard login method. The form we use to submit the username and password is the same as the standard mod_login module. I'd recommend trying to configure the mod_login module itself and see if you have the same issues with it. My guess is that you will run into the same issues. If so, you'll need to determine what is mis-configured with a separate SEF extension (if you're using one) or your .htaccess file (if you're using Joomla's built-in URLs).
Login and logout does not work when on Jomsocial pages
