First a side issue which I think is related:

I've been having a minor leak in my kitchen sink. The seal around the drain was bad and water would drip in the cabinet below. In the process of fixing it I think some debris such as small bits of old plumber's putty may have been rinsed down the drain.

Two days ago my kitchen sink stopped draining correctly. If run for more than a couple seconds it backs up dirty (almost black) water into my bathtub. The tub similarly doesn't drain now. The toilet and bathroom sink all drain fine. All fixtures are on the same level of a single story house with a basement.

Because it happened so close to my other repair and came on suddenly as opposed to gradually, I'm assuming they're related.

I've tried a chemical drain cleaner/block remover to no luck. My next step before I start figuring out how to pay for a plumber will be to snake the drain. A coworker will be lending me a snake that attaches to a hand drill for this purpose.

Where should I attempt to snake from? Should it be the kitchen drain or the tub drain?