I have been able to achieve what I need by changing the module here:

function youtube_uploader_get_settings_form() {
  $ret['youtube_uploader_publishing_options'] = array(
    '#type' => 'radios',
    '#title' => t('Default privacy options'),
    '#description' => t('More info on !here', array('!here' => l(t('YouTube support page'), 'http://support.google.com/youtube/bin/answer.py?hl=en&answer=157177'))),
    '#options' => array('public' => t('Public'), 'unlisted' => t('Unlisted'), 'private' => t('Private')),
    '#default_value' => strtolower(variable_get('youtube_uploader_publishing_options', 'unlisted')),
  );
  //Added: 'private' => t('Private') - being the added code.
}

How do I achieve this without hacking the module?