The more I think about this, the more I think I would exercise extreme caution.
Especially if the person knows personal information (like a family member who might just happen to know your social security number, for example.)
Once you have someone listed as an authorized user, I don't think it would be that hard for them to call up and have the company mail them a fresh card without your knowledge . . . particularly if they know enough of your personal details to get past security questions. Would this be fraud? Well, you authorized them to use the card, so from the CC companies perspective it is not fraud for them to obtain and use the card. That means you would be on the hook for any charges. Technically they are probably only supposed to let the primarily account holder change addresses, etc., but my experience is that, in practice, they will do quite a bit for an authorized user over the phone.