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When the user enters a page or post password WordPress sets wp_postpass_XXXX cookie:

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I installed Post Password Token plugin that sets this cookie with the following code:

function ppt_set_cookie($post_password) {
  global $token, $wp_version;
  
  setcookie(PPT_COOKIE.COOKIEHASH, $token, null, COOKIEPATH);
  $redirect_uri = 'http' . (is_ssl() ? 's' : '') . '://'.$_SERVER['SERVER_NAME'].$_SERVER['REQUEST_URI'];
  
  if (version_compare($wp_version, '3.3', '<=')) {
    // legacy cookie
    setcookie('wp-postpass_' . COOKIEHASH, $post_password, time() + 864000, COOKIEPATH);
    wp_redirect($redirect_uri);
  }
  else {
    // hashed cookie
    global $wp_hasher;

          if (empty($wp_hasher)) {
      require_once( ABSPATH . 'wp-includes/class-phpass.php' );
      // By default, use the portable hash from phpass
      $wp_hasher = new PasswordHash(8, true);
    }

    setcookie('wp-postpass_' . COOKIEHASH, $wp_hasher->HashPassword(stripslashes($post_password)), time() + 864000, COOKIEPATH);
    wp_safe_redirect($redirect_uri);
  }
  exit;
}

also it sets its own cookie wp-post-token_XXXX.

The plugin makes password protected posts and pages accessible with a direct link. It was not updated for a long time, but looks like it works somehow.

Links:

• Bypass a WordPress Password Protected Post or Page via a URL