An electronic device may have a housing in which an antenna is mounted. An antenna window may be mounted in the housing to allow radio-frequency signals to be transmitted from the antenna and to allow the antenna to receive radio-frequency signals. Near-field radiation limits may be satisfied by reducing transmit power when an external object is detected in the vicinity of the dielectric antenna window and the antenna. A capacitive proximity sensor may be used in detecting external objects in the vicinity of the antenna. The proximity sensor and the antenna may be formed using integral antenna resonating element and proximity sensor capacitor electrode structures. These structures may be formed from identical first and second patterned conductive layers on opposing sides of a dielectric substrate. A transceiver and proximity sensor may be coupled to the structures through respective high-pass and low-pass circuits.