I have two steam accounts, and I was not able to see anything related to a game marked private from my second account except when family sharing was enabled between the accounts. With family sharing on I could see all private games from my primary account on my secondary (including games which were not installed on the local system).
If you have family sharing on, hold off. Otherwise as far as I know it works as intended.
OP is not referring to private lobbies in a game, they are referring to marking a game private on Steam itself, a feature that hides from the world (and your friends) that you own or interact with a game.