Water backs up in adjacent sink only when garbage disposal is used
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).