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Social actors are often embedded in multiple social networks, and there is a growing interest in studying social systems from a multiplex network perspective. In this paper, we propose a mixed-effects model for cross-sectional multiplex network data that assumes dyads to be conditionally independent. Building on the uniplex p2 model, we incorporate dependencies between different network layers via cross-layer dyadic effects and through the covariance structure among the actor random effects. These cross-layer effects model the tendencies for ties between two actors and the ties to and from the same actor to be dependent across different relational dimensions. The model can also study the effect of actor and dyad covariates. As simulation-based goodness-of-fit analyses are common practice in applied network studies, we here propose goodness-of-fit measures for multiplex network analyses. We evaluate our choice of priors and the computational faithfulness and inferential properties of the proposed method through simulation. We illustrate the utility of the multiplex p2 model in a replication study of a toxic chemical policy network. An original study that reflects on gossip as perceived by gossip senders and gossip targets, and their differences in perspectives, based on data from 34 Hungarian elementary school classes, highlights the applicability of the proposed method. The proposed methodology is available in the R package multip2.