As of right now, when a customer responds to one of your SMS messages, this response will register under the Sent SMS metric. This will show in their profile with the response the customer has sent.
You can click on the timestamp to view additional details about the message.

Our feature roadmap includes a built-in SMS inbox, where you will be able to see all incoming SMS messages in one place. For now, you can view all Sent SMS events in Metrics > Sent SMS.