The correct request would be:

$url =
       "https://api.dailymotion.com/collection/xxx/videos?fields=private_id&private=true";

If you want to get more than 10 items, set limit like:

$url =
   "https://api.dailymotion.com/collection/xxx/videos?fields=private_id&private=true&limit=100";

100 is maximum you can get. Also learn the following information: https://developers.dailymotion.com/guides/browse-large-catalogs/