Some cards, yes, some cards, no. Chase United Mileage Plus cards? Yes ("Primary Cardmembers and one companion traveling on the same reservation will receive their first and second standard checked bags free each time they fly on United- or United Express-operated flights when they purchase their tickets with their MileagePlus Club or Presidential Plus Card"). Citi AAdvantage and Barclay Aviator cards? No (Same language in T&Cs of both cards; note absence of any purchase requirement: "For benefit to apply, the Account must be open 7 days prior to air travel AND, reservation must include the primary Cardmember's American Airlines AAdvantageŽ number 7 days prior to air travel.").

I can confirm through personal experience that flights not paid by an associated credit card on AA have received the free bag benefit.