When someone responds to a side conversation email there is a temporary notification for the assignee on the top right of the screen that lasts for about 10-15 seconds.
I am aware it is possible to configure a trigger based email notification to inform an assignee via email that there is a response to a side conversation email. That comes with an expense of agents being overwhelmed with number of emails.
I asked if it is possible to make a configuration change so that the side conversation email notifications appear in the "Updates to your tickets" of the agent's Home Dashboard and received a negative answer through Zendesk help center.
This would be an optimal way of notifying assignee about the responses to the side conversation emails for their tickets, alongside with other ticket updates.
I'd suggest including this in one of you upcoming improvements/ updates.