A received message should be visible in any SMS app, since they all use the system database for storing messages (Handcent, Chomp, Textra, etc all work this way). If you can't see the received messages when you open the 3rd party app please post again to tell us what apps so we can try to work it out (registration required, but it's free).
However, you will only receive notifications if you have enabled them in the app, so if you just install a new app but don't change the settings you may well only get the notifications from the old app when messages arrive. So if you want to use a different app, go into that app's settings and enable (and customise) notifications there. Then go into the stock app's settings and turn off its notifications so that you don't get notifications from both.
I hope that answers the question, but please let us know if not.