What do you mean by "privately owned"?
Is my body privately owned?
Do I own my body and labor?
What are capitalists?
In Capitalism you don't own your labor, capitalist does. Capitalist is person that owns a business in which other people work and are not equal members of the business.
Well, corporations are capitalist businesses, no one calls them capitalist businesses because no one wants to or dares calling it by real name as no one dares go against the system.
How is "capitalist" the proper name? And are you not aware of the thousands and potentially million of people who say the exact same thing you're saying? No, it's NOT no-one.