Communicate private health information? A lot of times they still use fax machines. Information can also be stored in a secured database where access is recorded and monitored. If needed, they can always pick up the phone and talk directly with a person if you need something. HIPAA is fairly specific about this.

An encrypted two-way radio, where only the two parties requiring the information would be on the call, that might be fine as long as you're careful to make sure someone standing nearby can't overhear. But that's not what NYC is building.