A social network is a blogging and chatting service which enables interaction with many people whether you know them or not. This is the public way to use Buzz.
However, there is also a private way to use it. Imagine you are in a small rock band and all members have a Google account. You can create a group in your contacts list for them and post private Buzzes to that group so only they can see and discuss it. In your rock band, you will be able to share and discuss your songs, instruments, rehearsals, next concert, drawings for the CD or whatever you want. If you want to involve someone out of the group, you can always poke him/her with a @reply and he/she will be able to interact in that specific post. Note that private posts aren't displayed on both your profile and the public stream accessible from the API.
There are many reasons and situations where private buzzing is required: Business, organizations of any type, education, family or groups of friends. It's up to you to invent your own private Buzz use.