Within Teamwork Desk there are some situations where it is expected behavior for Desk to not send your reply or notification.
For example, Desk won't send out an email if the email address of the customer is the same as the email address of the agent replying. However, Desk should still send a message to any CCs that may have been added though.
Desk also won't send notifications to this email address if the same.
For more information, see: Teamwork Desk account limits