Private shows are not a different gameplay mode. They are events that happen from time to time, depending on the Decorations you own and on Skill levels.

They take place when you see a red/pink/golden circle with an eye icon above a hired girl. Clicking on a girl while she is performing a private show will allow you to gain more income than usual.

Note: Last Private Show took place on 24 October, 2019 (Saya's Private Show)

Check wiki page for mode info: https://fapceo.miraheze.org/wiki/Private_Show