Contribution form no longer entering data into CiviMember for free memberships

CMS: WordPress 4.4.2
CiviCRM: 4.7.1

After upgrading from 4.6.6 to 4.7.1, when someone submits our contribution form for a free membership, the membership is not recorded in the individual contact record nor in the CiviMember area. What's worse is that these contacts have no record of their "contribution". So, they aren't displaying in "Find Contributions" either. Our memberships are free, so there is no paid transaction happening.

The contact makes it in the Contacts area and database OK but wondering why the connection to CiviMember and CiviContribute was severed.

I tested on the official sandbox and re-created the scenario and symptoms after posting on StackExchange about the problem.

Membership Types - http://wpmaster.demo.civicrm.org/wp-admin/admin.php?page=CiviCRM&q=civicrm%2Fadmin%2Fmember%2FmembershipType&reset=1

Free Membership Form - http://wpmaster.demo.civicrm.org/civicrm/?page=CiviCRM&q=civicrm/contribute/transact&reset=1&id=4

Inputted Contact - http://wpmaster.demo.civicrm.org/wp-admin/admin.php?page=CiviCRM&q=civicrm%2Fcontact%2Fview&reset=1&cid=207&key=6f2864fb86890dd1bf19ee6f183a5eda_9439&context=search

Even though everything appears to be configured correctly, the data that gets dumped in is missing several pieces (CiviContribute and CiviMember info).

Original StackExchange Question: http://civicrm.stackexchange.com/questions/9725/form-no-longer-entering-into-civimember-for-free-memberships