Very often proofs need to be reviewed by one group and you do not want their comments to be seen by other groups. For example, an agency may review a proof internally before it is shared with a client and internal comments shouldn't be visible to the client.
By default, stages are visible to all participants in the review. This means that the comments left by reviewers from those stages are visible to everyone and they are included in the email alerts and comment summaries.
Private stages are only visible to participants in those stages and users from your account who have edit rights on all items created in your account (so they have profiles of Supervisor and above or users with Custom profiles who have editing of other people's items enabled).
Comments left by participants from private stages won't appear in the proof email alerts and comment summaries requested by people who don't have the rights to view them.
To find out how to set up a proof with Automated Workflow, please take a look at the help article Adding and Managing Automated Workflows for Proofs.