An electronic device may be provided with metal coated dielectric structures that serve as electromagnetic interference shielding, antenna structures, or other metal structures. The metal coated dielectric structures may be formed form a sheet of polymer. Metal may be deposited on the sheet of polymer using a deposition tool and patterned following deposition or may be patterned during deposition. A dielectric sheet having patterned metal may be shaped into a desired shape using molding equipment or other equipment that applies heat and pressure to the dielectric sheet and patterned metal. Metal on a dielectric sheet may also be patterned after the dielectric sheet is formed into a desired shape. Metal may be formed on opposing sides of the dielectric sheet.