It's both. You can use various email addresses for the notifications you receive from the Coaches Console system as coach and for the messages you send to your own clients from the Coaches Console.
You can setup these email address for different types of messages in Communicate > Settings > see the email fields there.
The notifications you get as coach from the Coaches Console system are going to the default email address.
When you clients receive your Coaches Console messages they see your email address and the fact that the message comes from the Coaches Console on behalf of your email address.
Note: Your email address is something you obtain separately from your Coaches Console (we do not provide email addresses as part of our services).