Currently, email notifications for Offline Chat messages are sent to all chat licensed agents. The setting to turn off these notifications is at the individual licensee level (and within their profile settings) rather than controlled at an Administrator level. 

Business Impact: We strive to keep communication within Zendesk as a tool as much as possible. With many of our agents working different time zones, the email notifications are sent out to all and can be distracting or take extra time to investigate a matter that has already been solved.

The workaround: Either each individual support agent (or an administrator logged in as the support agent) needs to turn off the notification checkbox.

The suggested improvement: If the chat email notification settings could be controlled at an administrative level in Chat or Support, this would be an easier big picture configuration setting to change as needed.

Thank you for considering this request.