If you would like to be notified about any updates made to certain tickets that are not assigned to you or that you are not a part of, you can add yourself as a 'Watcher' by clicking on the star icon (or use the keyboard shortcut 'W') on the ticket page.
You can automate this by setting up an automation rule under Admin > Workflows > Automations > Ticket updates > New rule:
When an action performed by..
Involves any of these events:
Reply is sent
On tickets with these properties:
In tickets > if Requester email > is > firstname.lastname@example.org
Perform these actions:
Send 'Email to Agent' > customize your email
Alternatively, you can also add the corresponding agent as watcher under Actions.