There is generally one reason why people with double units have a clogged sink on both sides. The reason is that the drainpipe below where the two pipes meet is clogged.
Unclogging this pipe is actually not that difficult, so long as you have the right equipment. Be sure to have a plunger, bucket, wrench, and drain snake on hand. With your plunger at the ready, fill both your garbage disposal sink and kitchen sink with a quarter of water.
Be sure to plug in the garbage disposal sink after you fill it with the water, since this will help it form a seal. After that, you can begin to push down with your plunger and form a seal over the kitchen sink drain.
After about five plunges, you can check to see if hot water drains smoothly. If it drains down the kitchen sink, you can then unplug the garbage disposal and see if hot water flows freely.
If this first trick didn’t work, there are two more strategies to try.
First, you can inspect the trap underneath your kitchen sink, which is the “U” shaped bend in the pipe. Clear out the cabinet, place a bucket underneath the trap, and use a wrench to help you take the trap off.
If you see any debris in the pipe, just push it out with a coat hanger. You can then wash the pipe outside with a hose before reattaching it to the sink and testing if the clog is solved.
If this doesn’t work, you may need to use a plumber’s snake to push through the kitchen sink pipe. You can keep using the plumber’s snake until you have fully cleared out the debris.