There's nothing overly special about an interstate-style highway network. It's essentially just a large number of roads covering a long distance that are connected together, with emphasis on driving speed.
The private sector is more than capable of planning, building, connecting, maintaining, and operating roads of all types, widths, and lengths.
I don't know where you are, but in Canada the various governments end up having to resort to the private sector anyway for highway-related projects.
It's government interference that prevents there from being more privately-build and operated highways. That's exactly the problem I was talking about in my earlier comment. Government "jobs" are dead weight on the real economy, only inhibiting those who are actually productive.
It exists because the 401 wasn't built with sufficient long-term capacity, even after it was widened significantly.
I wouldn't really consider the 407 a true "private" highway, though. It's more a case of the private sector rescuing the government from a typical government-created debacle, and last I heard, the highway is now majority-owned by the CPP Investment Board (a Crown corporation).
Have you ever actually driven on the 407, by any chance? It's a much nicer experience than the 401, although having to pay a toll while also paying significant taxes can be annoying. Under a proper system, there'd only be usage tolls, and no taxes involved.