This paper examines the data-driven verification of structural balance of multiplex networks by utilizing the dataset generated by the bipartite consensus dynamics adopted by each node in the network. To this end, some necessary and sufficient conditions for the structural balance of multiplex signed networks from a graph-theoretic perspective have been provided. It is also shown that a multiplex signed network is structurally balanced if and only if its related compressed network is structurally balanced. Built on the proposed theoretical results and dynamic mode decomposition technique, a distributed data-driven approach is proposed for verification of the structural balance of large-scale multiplex signed networks without relying on the explicit knowledge on network topology.