The vinegar and baking soda, or its commercial equivalent, just foam up into the other sink and not down the pipes?
Head to your local hardware store and get a plumbers “snake”. This is a long, coiled, flexible pipe thingy that chucks into a drill. You feed the snake down the clogged drain and it pushes its way into and past the clog. It’s a very handy tool to have around (so is a drill, btw!). Alternatively, you can run a couple kettles’ worth of boiling water down your drain. Clogs are usually grease-based with vegetable matter added (or hair if it’s a bathroom). Dissolving the grease usually frees the clog. I also have a double sink. I have to plug the other sink’s drain, add a couple inches of water, and weight down the drain plug with a heavy jar in order to keep the boiling water from backing up.
Cover one drain with a damp cloth. Pour baking soda down the other. Then pour vinegar down. Immediately cover the drain with another damp cloth. Press down on the cloths in both drains. The force of foaming should push the clog through. If that doesn't work the clog is further down the line and may need to be rodded through from underneath.
I followed a tip I saw here on Hometalk and was stunned at how well it worked. My kitchen sink has one drain with garbage disposal, so I had to be careful with what I used. The suggestion was to bring a large pot of water to boiling. Then pour baking soda down into the drain, then drizzle some Dawn on top. Remove the boiling water off the heat and pour in a generous amount of white vinegar. Pour this down the drain. The boiling water will melt grease which gets dissolved by the Dawn and the reaction of the baking soda to the vinegar in the water churns everything like crazy. I figured baking soda and Dawn were cheap, so I was generous with both (same thing with the vinegar).
In your case, I would enlist the help of a partner. Get TWO large pots of water boiling. Add the baking soda & Dawn to BOTH sides of the sink, then each of you pour your pot of boiling vinegar water down each drain at the same time.
I hope this works for you, it sure worked for my clogged sink! Good luck!
This drain snake is super easy to use, and it's amazing the junk it pulls back up. (click for shopping link to it)