First, make sure that you are BCC'ing each client message to an email address you use. To learn how to do that, please see this related question.
Once you have that setup, you will be BCC'd on each of your client's message. If you don't see the messages coming through, Mowmo may still be processing it. It is not uncommon for a message to take 5-10 minutes before you see them delivered to you and your client's inboxes. If messages still are not being delivered after 5-10 minutes, contact support for further investigation.