How to let public access to the sitemap page?
As far as I can tell, this is permission based, so you need to give anonymous users the access site map permission.

From the module:

/**
 * Implements hook_permission().
 */
function site_map_permission() {
  return array(
    'access site map' =&gt; array(
      'title' =&gt; t('View published site map'),
    ),
  );
}

/**
 * Implements hook_menu().
 */
function site_map_menu() {
  $items['admin/config/search/sitemap'] = array(
    'title' =&gt; 'Site map',
    'description' =&gt; 'Control what should be displayed on the site map.',
    'page callback' =&gt; 'drupal_get_form',
    'page arguments' =&gt; array('site_map_admin_settings_form'),
    'access arguments' =&gt; array('administer site configuration'),
    'file' =&gt; 'site_map.admin.inc',
    'type' =&gt; MENU_NORMAL_ITEM,
  );
  $items['sitemap'] = array(
    'title' =&gt; 'Site map',
    'description' =&gt; 'Display a site map with RSS feeds.',
    'page callback' =&gt; 'site_map_page',
    'access arguments' =&gt; array('access site map'),
    'type' =&gt; MENU_SUGGESTED_ITEM,
  );

  return $items;
}