Check-Ins are only as useful as the communication and feedback that follows. Submitting your Check-In publicly vs privately will change who has viewing access, but feedback is always limited to managers of the user who submitted the Check-In.
To review your direct report's Check-Ins, go to My Team's Check-Ins and click Review Check-In on the specific team member.
Here you can see a specific team member's Check-In and leave feedback. Click into the text box below the answer that you want to leave feedback on.
Don't forget to save on the bottom of the page!