While classical liquid droplets are rounded, transitions have recently been discovered which render polyhedral water-suspended droplets of several oils. Yet, the mechanism of these transitions and the role of the droplets' interfacial curvature in inducing these transitions remain controversial. In particular, one of the two mechanisms suggested mandates a convex interface, in a view from the oil side. Here we show that oil-suspended water droplets can spontaneously assume polyhedral shapes, in spite of their concave interface. These results strongly support the alternative mechanism, where the faceting in both oil and water droplets is driven by the elasticity of a crystalline monolayer, known to self-assemble at the oil-water interface, independent of its curvature. The faceting transitions in the water droplets allow the fundamental elastic properties of two-dimensional matter to be probed, enable new strategies in faceted nanoparticle and nanoshell synthesis, and provide insight into the molecular mechanisms of morphogenesis.