In this study, we demonstrate nanostructured osmosis membranes that employ vapor-phase water transport to simultaneously achieve high rejection of solutes and a high permeability. The membranes consist of a hydrophobic, thermally conductive silica nanoparticle (SiNP) layer with tunable thickness supported by a hydrophilic track-etched membrane. The membrane permeability for water vapor is 1-2 orders of magnitude higher than hydrophobic microporous membranes used for osmotic distillation. This permeability is only mildly lower (similar to 3 times) than the equivalent water permeability of typical forward osmosis (FO) membranes. We also demonstrate the high selectivity of the SiNP membrane via urea permeation tests, where this membrane exhibits a 2-3 orders of magnitude lower urea permeability coefficient than a thin-film composite (TFC) FO membrane. Further measurements and theoretical analysis using the dusty-gas model suggest that membranes with a smaller SiNP layer thickness are capable of having comparable water fluxes to TFC FO membranes while maintaining higher selectivity. Our work demonstrates that thin, hydrophobic nanostructured membranes composed of thermally conductive materials have a great potential to significantly extend the applications of osmosis-driven processes to treat challenging water sources.