Why does dirty water come into my second sink when the garbage disposal is on?
Sounds like the drain after the disposal and connection to the other sink is partially blocked. I'd start with a drain cleaner that's designed for a partial blockage. In the kitchen, it's likely a buildup from cooking oils and grease that got into the drain. If that doesn't work, then you can remove the trap under your sink and remove as much crud from it as you can get. Note that the trap will be full of dirty water, so have a bucket under it when you remove it. If it's still giving you trouble, snake the line after the trap.