When I use the garbage disposal, the water (and grounds) comes up in the sink on the other side.

Drain pipe is blocked or partially blocked. Remove trap and snake the drain pipe.

Harold is bang on. You have a blockage. Take the trap apart and send the snake up in the wall. Put out about 6 feet and if that doesn't get it click on back.
The reason your disposal all movies water from tub to tub is that it acts like a centrifugal pump when you turn it on. Good luck, Tom