Not "private" as in "dedicated to voice traffic". Almost all modern telcos route their PSTN as packets on the same MPLS/IP or pure-IP network as internet traffic.
This network is usually "privately owned" by the telco in the USA, but is publicly owned or subsidized in many parts of the world.
But there is in general no "separate" PSTN other than last-mile copper in most of the world.
Yes, much traffic does traverse on IP now - but its private tightly controlled networks.