It depends. Most VoIP solutions that provide PSTN type service do use the public Internet. Many companies running VoIP between corporate sites have been using private networks for this service but because the quality of service on most public IP networks is so good today it's more practical to run VoIP on the public Internet. To address the concerns about security, you can run VoIP over a VPN between known locations that are having sensitive conversations. This is easily done as VPN hardware and software is very fast (was not true a few years ago) and does not affect voice quality.
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