I did this with my two nieces on a trip to Japan several years ago. They were a bit older - both over 18 - but still under the age of majority for our destination, which is 20. I didn't have the potential problems for our domestic flights but took a notarized letter from all parents, giving permission for me to take them out of the country and, most importantly, giving me limited power of attorney for health care decisions while out of the US. The only place either of the girls was quizzed by officials was upon re-entry to the US.
With a younger child, I recommend you do all of the above.