In the multiparty number-in-hand set disjointness problem, we have k players, with private inputs X1,...,Xk ? [n]. The players' goal is to check whether ?=1k X? = .... It is known that in the shared blackboard model of communication, set disjointness requires ?(n logk + k) bits of communication, and in the coordinator model, it requires ?(kn) bits. However, these two lower bounds require that the players' inputs can be highly correlated. We study the communication complexity of multiparty set disjointness under product distributions, and ask whether the problem becomes significantly easier, as it is known to become in the two-party case. Our main result is a nearly-tight bound of (n1-1/k + k) for both the shared blackboard model and the coordinator model. This shows that in the shared blackboard model, as the number of players grows, having independent inputs helps less and less; but in the coordinator model, when k is very large, having independent inputs makes the problem much easier. Both our upper and our lower bounds use new ideas, as the original techniques developed for the two-party case do not scale to more than two players.