Why is my Drupal ajax form not populating the radio buttons
I think you need to reorganize your code.
First you need two ['Private_list'] at the same function Form and at this same function form you need to declare the array variables for each private list.

$list1_options = array(......);
$list2_options = array(.......);


['Private_list1´]: these are the first choice the user need to do: here you use the ajax call back.

['Private list2']: these one have the choices the user need to pick according to the first selection on ['Private_list1´]: at this 'Private_list2' you use the wrapper prefix and suffix.

the ajax call back function should only look something like this most of the times: 

function example_change_callback($form, &amp;$form_state) {
  return $form['private_list2'];
}


Your basicly using the call back to identify wich form you need to update on event change.

I recommend you study this drupal dependent ajax example, dispate the example is using dropdown list, you should be able to aply it to radio buttons, just follow the same structure.

I hope i was hepfull, tell us later how it went...
Have a good coding!