An electronic device such as a portable electronic device has wireless communications circuitry. Antennas in the electronic device may be used in transmitting radio-frequency antenna signals. A coupler and antenna signal phase and magnitude measurement circuitry may be used to determine when external objects are in the vicinity of the antenna by making antenna impedance measurements. In-band and out-of-band phase and magnitude signal measurements may be made in determining whether external objects are present. Additional sensors such as motion sensors, light and heat sensors, acoustic and electrical sensors may produce data that can be combined with the proximity data gathered using the antenna-based proximity sensor. In response to detecting that an external object such as a user's body is within a given distance of the antenna, the electronic device may reduce transmit powers, switch antennas, steer a phased antenna array, switch communications protocols, or take other actions.