That functionality should be default for your Customer Notifications.
You can configure the email response in your Project Settings > Customer notifications > Templates.
It should look like below, and on the right you can see there is a variable dropdown where you can add "Message Content" to your templates.
This should push the comment message to the email notification for every comment as long as that custom notification is enabled.
You can do this with your internal notification scheme for Watchers as well.
Hope this helps,
Nick with Isos
To ensure that watchers get email notifications for comments on an issue.
Go to Jira Settings > Issues > Notification Schemes
Then hit "Notifications" on the scheme your project is using to edit the individual notifications.
It should look like this by default. But you can add watchers by drilling into the "Issue Commented" notification.
Sorry for my late response, it has been a busy week here!
For adding new custom notifications that are out of the box please refer to this documentation.
Notification are run through custom events in your workflows.
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