In increasingly many settings, particularly in neuroscience, data sets consist of multiple samples from a population of networks, in which a notion of vertex correspondence across networks is present. For example, in the case of neuroimaging data, fMRI data yields graphs whose vertices correspond to brain regions that are common across subjects. The behavior of these vertices can thus be sensibly compared across graphs. We consider the problem of estimating parameters of the network population distribution under this setting. In particular, we consider the case where the observed networks share a low-rank structure, but may differ in the noise structure on their edges. Our approach exploits this shared low-rank structure to denoise edge-level measurements of the observed networks and estimate the desired population-level parameters. We also explore the extent to which complexity of the edge-level error structure influences estimation and downstream inference.