Private shows can be initiated by clicking "Start private show" below the broadcasters camera window when viewing the site on desktop.
This will pop up an information box containing the broadcaster's "tokens per minute" cost, the minimum minutes, and will let you know if you will receive a recording of the show.
On mobile, the process is similar but there is no pop up. Information is provided next to the request button.
Note: Make sure to talk to the broadcaster first and have them agree before starting a private show.