FYI anyone else looking to implement Ryan's solution for authenticated ProcessWire pages: you can make the request via cURL and pass the cookies in along these lines:
$ch = curl_init();
curl_setopt($ch, CURLOPT_URL, "http://$_SERVER[HTTP_HOST]/path/to/content");
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_COOKIE, $_SERVER['HTTP_COOKIE']);
$content = curl_exec($ch);
BIG CAVEAT: set $config->sessionFingerprinting to FALSE, or the code above will log your users out.