Public contacts are contacts that appear in your own Personal contacts list and 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 share business personal contacts, marked with a purple icon, that are associated with your TeleConsole account and not with private personal contacts (that are saved on your mobile device or its cloud storage options).
You can make a contact public by editing it or when creating a new one. To do so toggle on the Share contact with everyone in the company . Toggle the setting off to make the contact private again - one that is only accessible to you.