Do I have to install a vent to add a bar sink in my basement?
Do I need a separate vent stack for my kitchen sink?
... in my state (VA), the allowable distance between a 1 1/2&quot; 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.

If your local codes allow them, the 2&quot; pipe may serve as a wet vent. According to the IPC an X size stack can handle Y number of DFU.
1.5&quot; : 1 DFU
2&quot;   : 4 DFU
2.5&quot; : 6 DFU
3&quot;   : 12 DFU
DFU or Drain Fixture Units, where the UPC allows for bar sinks to count as 1 and kitchen sinks to count as 2.
So long as these are the only two fixtures discharging into the 2&quot; pipe, you're within nominal code not to add a vent but you still need to check your local codes.