Plumbing – Order in size for fixture drains to enter the main stack

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.

905.2 Grade.

All vent and branch vent pipes shall be so graded and connected as to drain back to the drainage pipe by gravity.

(2006 International Plumbing Code).

This might prevent you from doing what you want, since your lower horizontal connection would have to slope toward the stack. Not sure if this is allowed or not, but it might be something to look into.