In this paper, we consider a distributed stochastic nonconvex optimization problem for multiagent systems. We propose a distributed stochastic gradient-tracking method with event-triggered communication. A group of agents cooperatively finds a critical point of the sum of local cost functions, which are smooth but not necessarily convex. We show that the proposed algorithm achieves a sublinear convergence rate by appropriately tuning the step size and the trigger threshold. Moreover, we show that agents can effectively solve a nonconvex optimization problem by the proposed event-triggered algorithm with less communication than by the existing time-triggered gradient-tracking algorithm. We confirm the validity of the proposed method by numerical experiments.