Public contacts are contacts that not only appear in your own Personal contacts list, but also in the Personal contacts list of every other user in the company. In other words, public contacts are shared by everyone in your company, while non-public personal contacts are only available to you.
You can only do this with business personal contacts that are marked with a purple icon (ones that are associated with your TeleConsole account) and not with private personal contacts (ones that are saved on your phone or its cloud storage options).
You can make a contact public by editing it or when creating a new one. To do so swith the Share contact with everyone in the company toggle to the on position. Toggle the setting off to make the contact private again - one that is only accessible to you.