It has been recognized that many complex dynamical systems in the real world require a description in terms of multiplex networks, where a set of common, mutually connected nodes belong to distinct network layers and play a different role in each layer. In spite of recent progress toward data based inference of single-layer networks, to reconstruct complex systems with a multiplex structure remains largely open. In this paper, we articulate a mean-field based maximum likelihood estimation framework to address this problem. In a concrete manner, we reconstruct a class of prototypical duplex network systems hosting two categories of spreading dynamics, and we show that the structures of both layers can be simultaneously reconstructed from time series data. In addition to validating the framework using empirical and synthetic duplex networks, we carry out a detailed analysis to elucidate the impacts of network and dynamics parameters on the reconstruction accuracy and the robustness.