Apply a penny-sized amount of low-abrasive polishing cream to a polishing cloth. Gently rub the metal in a swirl motion, taking care not put pressure on the engraving. If the scratches are still visible, repeat, and wipe the cream off with a clean section of the cloth. This may be all you need to do and is the best way to polish engraved metal because it has the least risk of causing damage.
Rub the engraved metal with a rogue cloth, available at jewelry supply retailers. This two-sided cloth has an rough side, usually red, and a polishing side, usually tan or white. Use the red side first, rubbing very gently to avoid damaging the engraving. Flip the cloth over and rub the metal with the polishing side until the surface is shiny and scratch-free.
Polish the engraved metal with a polishing wheel attached to a rotary tool if hand polishing does not remove the scratches. Set the rotary tool to its lowest rotation speed and quickly touch the spinning polishing wheel to low-abrasive polishing compound. Touch the edge of the wheel to the metal, moving the rotary tool in a swirl motion. Do not apply pressure. When the scratches are smooth, wipe the metal with a soft cloth.
Wash the polished engraved metal in warm water and liquid dish soap. If the piece has parts that should not be submerged, such as with a watch or a piece of metal attached to leather, rinse it with a damp cloth and wipe dry.