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Likelihood-free inference involves inferring parameter values given observed data and a simulator model. The simulator is computer code which takes parameters, performs stochastic calculations, and outputs simulated data. In this work, we view the simulator as a function whose inputs are (1) the parameters and (2) a vector of pseudo-random draws. We attempt to infer all these inputs conditional on the observations. This is challenging as the resulting posterior can be high dimensional and involve strong dependence. We approximate the posterior using normalizing flows, a flexible parametric family of densities. Training data is generated by likelihood-free importance sampling with a large bandwidth value epsilon, which makes the target similar to the prior. The training data is "distilled" by using it to train an updated normalizing flow. The process is iterated, using the updated flow as the importance sampling proposal, and slowly reducing epsilon so the target becomes closer to the posterior. Unlike most other likelihood-free methods, we avoid the need to reduce data to low dimensional summary statistics, and hence can achieve more accurate results. We illustrate our method in two challenging examples, on queuing and epidemiology.