How do I troubleshoot PrivateTempStore for session data?
I am updated one of my Drupal 7 modules for Drupal 8 and I discovered
  that I need to use PrivateTempStore rather than $_SESSION variables.


The Drupal 8 replacement for $_SESSION is not PrivateTempStore, it is the session object in the request:

Drupal 7:

function mymodule_session_counter_increment() {
  if (!isset($_SESSION['mymodule_count'])) {
    $_SESSION['mymodule_count'] = 0;
  }

  return $_SESSION['mymodule_count']++;
}


Drupal 8:

class MymoduleSessionCounter {
  function increment(Request $request) {
    $session = $request-&gt;getSession();
    $value = $session-&gt;get('mymodule_count', 0);
    $session-&gt;set('mymodule_count', $value + 1);

    return $value;
  }
}


See the change record: https://www.drupal.org/node/2380327

For more info see this topic: How do I store the values submitted in a form in the session?