In the absence of coalescence, coarsening of emulsions land foams) is controlled by molecular diffusion of the dispersed-phase species from one emulsion droplet (or foam bubble) to another. Previous studies of dilute emulsions have shown how the osmotic pressure of a trapped species within droplets can overcome the Laplace pressure differences that drive coarsening, and "osmotically stabilize" the emulsion. Webster and Cates (Langmuir 1998, 14, 2068-2079) gave rigorous criteria for osmotic stabilization of mono- and polydisperse emulsions, in the dilute regime. We consider here whether analogous criteria exist for the osmotic stabilization of mono- and polydisperse concentrated emulsions and foams. We argue that in such systems the pressure differences driving coarsening are small compared to the mean Laplace pressure. This is confirmed for a monodisperse 2D model, for which an exact calculation gives the pressure in bubble i as P-i = P + Pi + P-i(G), with P the atmospheric pressure, IT the osmotic pressure, and P-i(G) a "geometric pressure" that reduces to the Laplace pressure only for a spherical bubble, and depends much less strongly on bubble deformation than the Laplace pressure itself. In fact, for Princen's 2D emulsion model, P-i(G) is only 5% larger in the dry limit than the dilute limit. We conclude that osmotic stabilization of dense systems typically requires a pressure of trapped molecules in each droplet that is comparable to the Laplace pressures the same droplets would have if they were spherical, as opposed to the much larger Laplace pressures actually present in the system. We study the coarsening of foams and dense emulsions when there is an insufficient amount of the trapped species present. Various rate-limiting mechanisms are considered, and their domain of applicability and associated droplet growth rates discussed. In a concentrated foam or emulsion, a finite yield threshold for droplet rearrangement among stable droplets may be enough to prevent coarsening of the remainder.