Cannot find to make this in their LiveBroadcasts: insert documentation
Any idea which parameter need add to body?
As YouTube Live Streaming API LiveBroadcasts: insert endpoint doesn't seem to support this feature. Maybe you could manually after your livestream proceed with YouTube Data API v3 Videos: update endpoint by changing the privacy status thanks to snippet.privacyStatus. If you are interested in detecting automatically when your livestream ended, then this section of the YouTube Live Streaming API documentation may interest you.