A not logged in user can set a name. However a value is not sent in Drupal.chatroom.postMessage in chatroom.js. Every time an empty string is set before:
var anonNameText = '';
It should be replaced with something like:
var anonNameText = $('#edit-chatroom-anon-name-' + chat.cid).val();

Comments

Version:7.x-2.0» 7.x-2.x-dev

thanks for the report.