The Private Objectives are added mainly when the information in OKRs are confidential to the user. Hence, the private OKR is not visible to other users.
The following table explains the functionality that you can or cannot apply on a private objective.
Assigning the Owner
Restricted as an individual OKR.
View and Edit
Only from the My OKRs page by the owner.
Search through Explore
Allows to edit details
Convert into a Public Objective
Yes. While editing a private objective you can convert it to a public objective.
Does not allow cloning it to create another private objective.
Yes. You can export with the objective the My OKR list.