What does the private/public notification label next to names on family search mean?

In general, a Private Person label means that the person is living, and this only you can see that profile -- multiple users might have distinct profiles for a living person, but each user can only see their own copy. (It's possible to have a deceased confidential person who is also designated as a Private Person, but that is rare.)

A Public Person label is for a deceased person; such profiles are visible to everyone and thus are naturally shared among all users of Family Tree.

A third possibility is that a profile can be labeled with the name of a Family Group Tree. This is available only to members of a Family Group that has a Family Group Tree. In this car, group members can share a single profile for a living person.