Softer materials, such as opal, require treatment more like that accorded cut glass, and soft abrasive powders, such as pumice, suffice to polish them &should be used with water or oil to give it a creamy consistency. It should be backed by laps of different materials for different purposes. You can use a domestic drill, but by the time you've paid for the special abraisives you may as well pay the jeweller!
I would try jeweller's rouge on a cloth. This is designed to polish up gilver & gold jewellery removing the fine scratches of everyday wear & tear. Polish in a linear direction not round & round. Then keep your jewellery seperately boxed when you're not wearing it to prevent further scratches & keep opal, pearl,amber, mother of pearl shell out of sunlight because they will lose their shine & colour. Ditto for doing housework & washing up - take off your rings unless they're diamond!.