We've read that the best silver polish is white non-gel toothpaste. Dab some on your finger, and rub into the tarnish.
Another eco-friendly option is to make a paste of baking soda and water, scoop it onto a clean, damp sponge, and rub the paste into the silver. Rinse with hot water and polish dry with a soft, clean cloth. For badly tarnished silver, leave the baking soda paste on the silver for an hour or so, and then clean it off with the sponge and hot water.
But we admit we haven't actually tried these. If you have, please comment below!