I've done every possible combination with the added bonus of my last name being different from my kids and that we don't share the same nationality (they are American, I am not). Though I always have passports, birth certificates and a notarized letter from dad with a copy of his passport or his license (or he has one from me) we have NEVER been asked for any documentation. We travel a lot. I actually think this is scary...
Getting a notarized letter takes 5 minutes at any UPS store. I keep getting one for every trip because the one time I won't have it all hell will break loose