They don't mirror your mail, you receive a pushed mail on the msgpush account telling you new mail has arrived.
You still have to read it on your regular account
These is a background notification service any iPhone app can use that can show a text message on the screen and update a counter (aka "badge") on the icon of an app that isn't running.
Push notification in iPhone OS 3.x is not to be confused with Exchange or ActiveSync so-called Push email.