The flexibility of mixed fluid membranes is examined in terms of the so-called hat model that is based on local curvature fluctuations. For monolayers, the hat model reproduces the increase of the flexibility obtained previously with a continuum model. In bilayers, the bending frustration of the constituent monolayers causes equal molecules to avoid and unequal molecules to seek each other when they are in opposite monolayers. This local interaction across the middle surface of the bilayer leads to an additional increase of bilayer flexibility. The cooperative effect of local bending frustration in partially polymerized bilayers may result in the segregation of polymer chains belonging to opposite monolayers. Possible consequences of the segregation are a nearly divergent flexibility, a pronounced roughness of the bilayer, and budding of bilayer vesicles.