We develop a scalable method to estimate the parameters in models of very large binary network datasets. Maximum likelihood estimates are generally impossible to obtain because the full likelihood involves an intractable high dimensional integral. Also, full-likelihood Bayesian estimation is impractical for very large datasets as the MCMC algorithm is very slow. We propose a triadic composite likelihood estimation method for exchangeable latent Gaussian network models, and extend it to q-node composite likelihood estimation for other exchangeable and non-exchangeable models. The maximum composite likelihood estimates are obtained by optimizing the composite likelihood using a stochastic gradient-based algorithm, where the gradients are approximated using Monte Carlo samples. For networks of moderate size, we show via simulations that composite likelihood estimation provides estimates as accurate as those provided by fully Bayesian estimation using MCMC. For very large datasets, fully Bayesian estimation is impractical, but composite likelihood estimation is feasible as its computational cost is essentially constant as a function of the network size.