Once your tournament has been Published for the first time by default registration will be opened so that players may start to join your tournament. A tournament being set to Private or Public does not impact whether or not registration is open or closed.
To manage registration's open or closed status you will return to the Manage section of your tournament page, then navigate to Setup - Settings. A new toggle button will be available in the Required Fields section that will allow you to close registration. You can also reopen registration from here, however once the tournament has started players will not be able to register regardless of the status of this button
For more information about setting up and managing your tournament click here.