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A common address is all information a CRA needs to identify the AU's credit profile.
My mother added me as an AU to her Chase card (I've since been removed) and did not provide my SSN. She lives in NY and I live in CA. It showed up on my CR the next month.
Amex and City both ask for SSN, I know this as after 9 months in the country I had 9 years of credit history and a fico @720 and a credit line (ink AU and auto loan) for well over 100K.