How can occasional agents get notifications for tickets assigned to them without signing into Freshdesk and send replies to customers directly through email?

Please navigate to Admin --> Workflows --> Email notification --> Agent Notification --> and turn on the notification for "New Ticket Created". Once this is done, please click on "Edit" corresponding to this email notification, and under the "Notify Agents" section you could have to add the occasional agent. The occasional agents can send replies from their mailbox.