Broadcasters can choose to allow private show recordings in their "Settings & Privacy".
To find out if a broadcaster allows this, initiate a private show and before confirming, it will tell you if they have this feature enabled or not.
After the private show, if the broadcaster allows private show recordings, you can find it by clicking My Collection below your username on the home page.
My Collection is also available on mobile. You can find it by going to the homepage on the mobile site, then opening the menu on the left:
Private show recordings allowed by the broadcaster that last 30 seconds or longer will be saved in your collection.