I think there's a kind of ideology behind the phrase that personifies companies, giving them "rights" that become even more important than individual human rights. And there's the weird idea that companies can do whatever they want, despite the mountains of government regulation of corporate behavior.

Almost every use of "private company" is in contrast to a publicly traded company. Except when people start talking about "rights", and then they're apparently using it in a different way, which I find very odd and ideological.