You don't have a real trap. That tee and all the piping to the left and right make up a sort of trap, and will be full of water at all times. I can almost guarantee that both sinks drain very slowly. The tee does not look like a baffle tee, so when the disposer runs, a lot of waste will probably be directed into the other bowl.
The root cause is that the outlet to the wall is too high, and should be lowered by opening the wall up.