What is a private building?
Private buildings are buildings that are not owned by you -- you can't set the wages or working conditions, and you don't receive any profits from them aside from a monthly rent. You are expected to make sure it's supplied with sufficient manpower and resources; if it fails to make a profit, it gets torn down.
You can get offers for private buildings if you have a Stock Exchange and either an economy minister or a foreign minister (I think; not sure about the ministers)