I also had some questions surrounding AU's, but thanks to the help of the community + further research I learned a lot from it.
Basically, AU's must give a social security number. This is good because it reports it to their credit but also holds them accountable for their purchases. However, as you said, they could go on a crazy spree on the card and then it would report the utilization and balance to your credit rating as well. American Express allows the primary card holder to set limits for AU's, such as spending limits and can also lock the card. Their credit report will not get 'linked' to yours, BUT whatever they decide to do on the AMEX will report to yours as well. I recommend setting a limit on their card, that way you wouldn't have to worry about some of these things. Hope this helps!