Consider a stochastic system of multiple subsystems, each subsystem having binary ("0" or "1") output. The full system may have general binary or nonbinary (e. g., Gaussian) output. Such systems are widely encountered in practice, and include engineering systems for reliability, communications, and sensor networks, the collection of patients in a clinical trial, and Internet-based control systems. This paper considers the identification of parameters for such systems for general structural relationships between the subsystems and the full system. Maximum likelihood estimation (MLE) is used to estimate the mean output for the full system and the "success" probabilities for the subsystems. We present formal conditions for the convergence of the MLEs to the true full system and subsystem values as well as results on the asymptotic distributions for the MLEs. The MLE approach is well suited to providing asymptotic or finite-sample confidence bounds through the use of Fisher information or bootstrap Monte Carlo-based sampling.