An electronic device may have a display. The display may have a central active area and a peripheral inactive area. A cover layer such as a layer of cover glass may be used to cover the display. The inner surface of the cover layer in the inactive area may be coated with an opaque masking layer such as black ink. Light for an ambient light sensor or other component may pass through an opening in the black ink. A layer of ink or other structures may be mounted within the opening to serve as a filter structure. A light diffusing structure such as a clear member with a textured surface may be used to diffuse light that is passing through the opening in the black ink and the filter to the light sensor. Integral filter-diffuser structures and structures with concave surfaces for focusing incoming light onto the light sensor may be used.