Customers can view the history of tickets if they have access to your customer portal.
They could log into your portal using the email address used to raise the tickets and view the status of all the tickets raised. You would be able to determine who could view the tickets by changing the permission to "Logged in Users" or to "Everyone" with a public ticket URL in Admin -> Channels -> Portals -> Settings.
Customers would also receive an email notification with the ticket details if you have enabled the Requester notifications under Admin -> Workflows -> Email Notifications.