Diffusion through semipermeable interfaces has a wide range of applications, including molecular transport through biological membranes, reverse osmosis, synaptic receptor trafficking, and drug delivery. In this talk I develop a probabilistic model of interfacial diffusion based on snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either side (U or V) of a semipermeable interface S. Each round is killed (absorbed) at the interface when its Brownian local time exceeds a random threshold. (The local time is a Brownian functional that keeps track of the amount of contact between a diffusing particle and a boundary surface.) A new round is then immediately started in U with probability p or V with probability 1-p. If p ≠ 1/2 (directed transport), then there is a discontinuity in the chemical potential across the interface. We show that the probability density for snapping out BM satisfies a renewal equation that relates the full density to the probability densities of partially reflected BM on either side of the interface. In the case of an exponentially distributed local time threshold with rate k, the solution of the renewal equation satisfies the classical diffusion equation for a semi-permeable membrane with constant permeability k/2. On the other hand, if the threshold distribution is non-exponential, then the resulting permeability is a time-dependent function that tends to be heavy-tailed. We end by considering the stochastic thermodynamics of diffusion across a semipermeable interface.