We study nonparametric estimation problems for discrete-time stochastic networks of Geom^X/G/∞ queues. We assume that we are only able to observe the external arrival and external departure processes at the nodes over a stretch of time. Based on such incomplete information of the system, we aim to construct estimators for the unknown general service time distributions at the nodes without imposing any parametric condition. We propose two different estimation approaches. The first approach is based on the construction of a so-called sequence of differences, and a crucial relation between the expected number of external departures at a node and specific sojourn time distributions in the network. The second approach directly utilizes the structure of the cross-covariance functions between external arrival and departure processes at the nodes. Both methods lead to deconvolution problems which we solve explicitly. A detailed simulation study illustrates the numerical performances of our estimators and shows their advantages and disadvantages.