What would cause a kitchen sink to drain very slowly

To answer the specific question of whether you need a separate vent for this sink: in my state (VA), the allowable distance between a 1 1/2" drain trap (standard size for a kitchen sink) and its vent is 6 feet. If your trap is farther from the vent than that, then you should consider adding a new vent or an air admittance valve.

That said, too much distance from the vent won't cause your sink to drain slowly. The problem with it being too far away is that the column of water flowing away from the trap is large and heavy enough that it has the potential to suck the water out of your trap as it leaves. And along with this sucking comes the gurgling sounds mentioned in the other answers here.

EDIT: To be clear, your sink doesn't have to be 6 feet from the vent stack, i.e. the main vent pipe that goes through the roof. Your DWV system may have other vent pipes that converge in the attic at the main stack.

There is a very simple answer to your question. Your dishwasher is draining into the side drain adapter on your garbage disposer. this was a common practice for many years. All the water draining from your dishwasher is being pumped into the upper basin side of the disposer. Solids from the dishwasher are blocking the drain in the disposer, so when you turn on the disposer, it grinds the waste and drains the water. This is actually the way it is suppose to work. Current codes require a separate trap for the dishwasher and not plumbed into the disposer. If you look at the side of your disposer, you will see the hose from the washer attached there. Only fix is to install a new separate trap and drain for the washer. Good luck.