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We study the permeability and selectivity (`permselectivity') of model membranes made of polydisperse polymer networks for molecular penetrant transport, using coarse-grained, implicit-solvent computer simulations. The permeability $\mathcal P$ is determined on the linear-response level using the solution-diffusion model, $\mathcal P = {\mathcal K}D_\text{in}$, $\textit{i.e.}$, by calculating the equilibrium penetrant partition ratio $\mathcal K$ and penetrant diffusivity $D_\text{in}$ inside the membrane. We vary two key parameters, namely the monomer-monomer interaction, which controls the degree of swelling and collapse of the network, and the monomer-penetrant interaction, which tunes the penetrant uptake and microscopic energy landscape for diffusive transport. The results for the partition ratio $\mathcal K$ cover four orders of magnitude and are non-monotonic versus the parameters, which is well interpreted by a second-order virial expansion of the free energy of transferring one penetrant from bulk into the polymeric medium. We find that the penetrant diffusivity $D_\text{in}$ in the polydisperse networks, in contrast to highly ordered membrane structures, exhibits relatively simple exponential decays and obeys well-known free-volume and Kramers' escape scaling laws. The eventually resulting permeability $\mathcal P$ thus resembles the qualitative functional behavior (including maximization and minimization) of the partitioning. However, partitioning and diffusion are anti-correlated, yielding large quantitative cancellations, controlled and fine-tuned by the network density and interactions as rationalized by our scaling laws. As a consequence, we finally demonstrate that even small changes of penetrant-network interactions, $\textit{e.g.}$, by half a $k_\text{B}T$, modify the permselectivity of the membrane by almost one order of magnitude.