A wearable electronic device such as an earbud, wristwatch, or other device may be provided with a skin sensor. The skin sensor may use optical measurements to detect the presence of skin adjacent to the electronic device. The sensor may have first and second light-emitting devices such as infrared devices that emit light at respective first and second infrared light wavelengths. Reflected light is monitored by a photodetector. Control circuitry can initiate or pause audio playback or take other actions in response to determining from the reflected light measurements that skin is present. The sensor may have a thin-film interference filter or other optical structure that overlaps the first and second light-emitting devices to narrow the angular spread of light emitted from the skin sensor. This reduces tilt sensitivity and helps enhance skin sensor accuracy.