What you are doing is absolutely correct. If you consider TripIt as the hub of your data, you can download to both your iPhone and iPad as long as both devices are signed into the same TripIt account. When I hear people having issues with this, more often than not, users have more than one TripIt account without realizing it. The TripIt login screen always prompts you to create an account which can sometimes trip (no pun intended) users. I would recommend you sign out from the Flight Update Prefs screen and sign back in from each device. This will insure that you are signed into the same account.
Let us know how you make out.