I have a generic garbage disposal, and if I run it with water it runs fine (it spins fine). However once it spins through the water and starts spinning "air" something akin to swamp gas (looks like faint smoke, and smells like rotting) starts coming up out of the drain on the other side. I don't think it's actually swamp gas coming up through, since there is still water in the P trap.
Swamp gas is pretty close actually. Most disposers have quite the bacteria culture growing inside it's mechanism. What you are seeing and smelling is an aerosolized portion of this culture ejected from the disposer by the spinning action. It can't go down the drain because it is blocked by the water in the trap. It doesn't normally come up through the disposer inlet because the air is cool and humid and dense. The typical rubber baffle also restricts passage.
Once the disposal is running without water, it is acting like a rather inefficient pump, pushing the smelly air and moisture down it's outlet. The easiest path for gas to follow from there is out the other drain.
Often, occasionally grinding up citrus fruit is adequate to keep odors in check.The acidity does slow down the bacteria, but the citrus is mainly masking the odor. Pouring a box of baking soda into the disposal will absorb some odor, and the basic (opposite of acid) environment also slows down bacteria. In addition, the powder can be left to sit for a while before it is washed away.
The only sure way to eliminate the bacteria colony is to block the outlet by removing a section of pipe and obstructing the opening with a secure plug. Fill the disposal with a strong chlorine bleach solution and let it soak for a few hours. Place a bucket under the plugged drain before removing the plug to catch the solution.
Even if you completely eliminate the colony with bleach, introducing more kitchen waste will cause another colony to establish itself eventually. It does help to only run the disposer when water is running. It helps prevent the bacteria from being aerosolized. You should do this anyway, running the disposal without water can cause it to overheat. This additional heat also enhances the smell that is emitted.