A model for the spreading of online information or ‘memes’ on multiplex networks is introduced and analyzed using branching-process methods. The model generalizes that of (Gleeson et al 2016 Phys. Rev. X) in two ways. First, even for a monoplex (single-layer) network, the model is defined for any specific network defined by its adjacency matrix, instead of being restricted to an ensemble of random networks. Second, a multiplex version of the model is introduced to capture the behavior of users who post information from one social media platform to another. In both cases the branching process analysis demonstrates that the dynamical system is, in the limit of low innovation, poised near a critical point, which is known to lead to heavy-tailed distributions of meme popularity similar to those observed in empirical data.