I'm not really confident about my answer to this, but since no one else has responded, I'll post what I think and someone more knowledgable will gleefully correct it :)
Components provide methods for saving and restoring state. Between requests, that state information is stored somewhere, but not in the component itself. If you choose server-side state saving, it's stored in your session. If you choose client-side state saving, it's stored in hidden fields on your response. So if you're using client-side state saving, you can have a different view of the component on every page. If you're using server-side state saving, you wouldn't (might be wrong about this -- you'd have to investigate the source and see). I think it's also possible to implement your own server-side state manager to maintain multiple views for a component, but that's speculation.
