Public contacts are contacts that not only appear in your own Personal contacts, but also in the Personal contacts of every other user in the company. In other words, public contacts are shared by everyone in your company, while non-public contacts are only available to you.
You can only do this with business personal contacts that are marked in a purple icon (ones that are associated with your Teleconsole account) and not with private personal contacts (ones that are saved on your phone).
To make a contact public simply open the contact's details and toggle the contact's status to be Set (to) Public mode.
Toggle the setting off to make the contact private again - one that is only accessible to you.