Please see Configuring the settings and make sure that step 5 has been followed for Public comment added and step 4 for Public comment edited Rules. These steps uncheck the Enable option for the built-in Jira Service Desk notification rules. 

If the Enable option is not switched off the built-in Jira Service Desk notification will run alongside the notification email generated by the app and will result in two emails being set out to customers. 

To double-check, open the Jira Service Desk Customer Notifications settings for the project and ensure that Public comment added and Public comment edited are disabled: