"When we talk about the private/public distinction, we sometimes mean to refer to what might better be called the privacy/publicity distinction ... The private/public distinction is often used, however, to mean something analytically different. It refers to a distinction in the legitimate locus of power that might best be called the private/governmental distinction" - Professor Peter M. Shane https://www.huffingtonpost.com/peter-m-shane/privatization-is-not-privacy_b_5132844.html
I'm not convinced "private" is the right word here either. As with fudging together privacy and privatization, "private" in this context is used to fudge together the "personal" and the "commercial", and manipulate people into thinking they and the corporations have interests in common in protecting themselves against the "governmental". In the current system it's often very important for people to use the governmental to protect the personal against the commercial.