Do I need a separate vent stack for my kitchen sink?
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.