CartOperationServices sends out a checkout completion notification when a cart finishes going through checkout. NotificationServices listens for this message in com.broadleafcommerce.notification.service.messaging.NotificationOrderConfirmationMessageListener. Out-of-the-box, the message listener looks for a handler to send the notification. If you want to bypass this logic, you can override methods in that class to do what you want (i.e. not send any notification) - overriding either processMessage() or validateShouldSendNotification() should work for this.
Moreover, returning false from validateShouldSendNotification() would disable all all notifications regarding a successful checkout and new order creation.