Permeation of water across the membrane/vapor and membrane/liquid-water interfaces of Nafion is studied using nonequilibrium molecular dynamics (NEMD) simulations, providing direct calculations of mass-transfer resistance. Water mass transfer within one nanometer of the vapor interface is shown to be 2 orders of magnitude slower than at any other point within the membrane, in qualitative agreement with permeation experiments. This interfacial resistance is much stronger than the resistance suggested by prior simulation work calculating self-diffusivity near the interface. The key difference between the prior approach and the NEMD approach is that the NEMD approach implicitly incorporates changes in solubility in the direction normal to the interface. Water is shown to be very insoluble near the vapor interface, which is rich in hydrophobic perfluorocarbon chains, in agreement with advancing contact angle experiments. Hydrophilic side chains are buried beneath this hydrophobic layer and aligned toward the interior of the membrane. Hydrophilic pores are not exposed to the vapor interface as proposed in prior theoretical work. At the membrane/liquid-water interface, highly swollen polymer chains extend into the liquid-water phase, forming a nanoscopically rough interface that is consistent with atomic force microscopy experiments. In these swollen conformations, hydrophilic side chains are exposed to the liquid-water phase, suggesting that the interface is hydrophilic, in agreement with receding contact angle experiments. The mass-transfer resistance of this interface is negligible compared to that of the bulk, in qualitative agreement with permeation experiments. The water activity at the vapor and liquid-water interfaces are nearly the same, yet large conformational and transport differences are observed, consistent with a mass-transfer-based understanding of Schroeder's paradox for Nafion.