First let me say that if you use a tumbler to work-harden and clean your jewelry pieces, that you should never mix metals in the same tumbling batch. Each batch needs to be one pure metal such as all copper or all silver or all brass (this means no mixed metal pieces such as one made with silver and copper, etc.) Stainless steel shot needs to be cleaned between each and every run.
Now let's look at a few reasons why the shot turns black. When using a new rubber barrel (even if you cleaned it well before the first use) until it is 'broken in', often the rubber on the inside comes off during the first 3 or 4 runs, leaving an undesirable coating on both the shot and your jewelry (especially if it is one of the inexpensive barrels). If you were running a batch that included mixed metals, copper and brass will tumble off a bit of residue that will adhere to all of the material in the batch, including the shot. If you used a dish detergent with a citrus additive like 'lemon fresh' or something, the citrus will turn all of the metal black. If you use city water, check to see if it contains additives that could be reacting with your metals and if so, use purchased water in a jug.
Make a thick mixture of baking soda and water and run it with the shot for 2 or 3 hours; remove the shot, rinse everything clear and if still blackened a bit, repeat.
Now here are a couple of methods to clean the shot.
Run the shot in clear water with a good dose of pure (no citrus) Dawn dishwashing detergent with grease cutter.
Using fresh water each time, repeat either cleaning method chosen, as many times as necessary to get rid of the collected grime.
Answer contributed by Dale "Cougar" Armstrong