An electronic device may be provided with antenna structures. Circuitry such as radio-frequency transceiver circuitry and impedance matching and filter circuitry may be implemented using one or more circuit components and embedded within an insulator to form packaged circuitry. The insulator may be formed from multiple layers of printed circuit board material or from plastic molded onto a printed circuit board substrate over the circuitry. A metal shield layer may be interposed between the packaged circuitry and the antenna structures. The metal shield layer may be mounted on the surface of the packaged circuitry using a layer of adhesive. A layer of polymer may be interposed between the layer of adhesive and the metal shielding layer. The metal shield layer may have an opening through which conductive paths may pass to couple the packaged circuitry to antenna terminals on the antenna structures.