Before revision:  In past work on this and also the more general mixture-of-products problem (X_i independent conditional on U, but not necessarily iid), a barrier at m^{O(k^2) a sample of size \exp(k\log k) (with m=2k).
After revision:  We give an algorithm for identifying a k-mixture using samples of m=2k iid binary random variables using a sample of size \exp(k\log k) (with m=2k).