Electronic devices may be provided with displays that have polarizers. A polarizer may have a polymer layer that has a portion covered with a dichroic dye to form a polarized region and a portion that is free of dichroic dye to form an unpolarized region. The unpolarized region may be formed by masking the polymer layer during a dye coating process. Masks may be formed from polymers. The shape of the mask may define the shape of the unpolarized region. The mask may be left in place within the polarizer or may be removed from the polarizer during fabrication. Unpolarized regions may also be formed by light bleaching, chemical bleaching, and material removal techniques. Bleached areas may be chemically stabilized. A moisture barrier layer may be incorporated into the polarizer to help prevent the unpolarized region from reverting to a polarized state.