I have adjacent sink basins in the kitchen with the garbage disposal on the right hand side. Individually, both basins drain quickly when the garbage disposal is not used, even when filled with water. However, when using the right side basin with the disposal going, it drains as quickly, but water will almost immediately backup into the left side basin until it reaches the same level as the diminishing water level in the disposal basin. Both drains then subsequently drain leaving food debris in the left basin. Is this a case of there being a partial clog some distance down the drain pipe? It may be implied that the volume capacity of the drain pipe is easily sufficient for water draining from a full sink without the "push" of a garbage disposal, but once the "push" factor is added with the garbage disposal going, the flow into the drainage pipe is increased sufficiently that now the capacity of the drain pipe (with a partial clog far down it) may not be enough to handle it. The added water in the drain pipe has no place to go but into the adjacent sink basin. Is this a correct interpretation of the problem or is there something more obvious going on?
You probably have a bit of a clog in your trap or a bit further down.
Likely, your plumbing looks like this:
The disposal is trying to pump water down the pipe much faster than it would passively flow. This means there is enough water getting into the shared pipe that it backs up into the sink on the right in the picture, bringing disposed food with it. Basically, it's probably exactly what you thought it was.
Try removing and cleaning the trap. If the problem still occurs, start trying some other drain cleaning techniques (snake, chemical clear, etc).
We had the SAME problem with our garbage disposal, sending food and whatever else was in the pipes , to our tub area ! Because the pipes were configured so close as this seems to be with you , anytime that the garbage disposal would run , it would send all this mess in the tub. We found out that later down the pipes, it was beginning to get clogged due to us not cleaning the pipes with a good drain cleaner , then Boiling hot water. ( cuts grease, etc). Once we did this , we had no more problems with the garbage disposal . Hope this helps !