If two quantum players at a nonlocal game $G$ achieve a superclassical score, then their measurement outcomes must be at least partially random from the perspective of any third player.  This is the basis for device-independent quantum cryptography.  In this paper we address a related question: does a superclassical score at $G$ guarantee that one player has created randomness from the perspective of the other player?  We show that for complete-support games, the answer is yes: even if the second player is given the first player's input at the conclusion of the game, he cannot perfectly recover her output.  Thus some amount of \textit{local} randomness (i.e., randomness possessed by only one player) is always obtained when randomness is certified from nonlocal games with quantum strategies. This is in contrast to non-signaling game strategies, which may produce global randomness without any local randomness. We discuss potential implications for cryptographic protocols between mistrustful parties.