Adding insulation in the wall DOES in fact reduce the sound transmission, and by quite a bit. Contrary to hj's assertion, most of the sound transfer between rooms is via air, not the framing. Air-sealing around electric boxes with an always-flexible acoustic sealant caulk (
Tremco makes one of the better acoustic sealants) or "Windows & doors" low expansion can-foam counts, as does caulking the studwall plates, and wall gypsum where it meets the floor or ceiling with acoustic sealant.
Then, adding well-fitted batts or blown cellulose/fiberglass cavity fill in a standard 2x4 partition wall provides a reduction of about 35-40 STC points(!). Don't forget to block the ceiling & floor joist paths too, if you have access- when you take the wall's transmission paths down 40 STC points the ceiling or floor joist bays can start to dominate. Recessed lights into open joist bays are an acoustic super-highway, but even empty joist bays and unsealed electrical/plumbing penetrations in the ceilings & floors is more than a mere farm-road path for sound.
The wallboard layers are effectively a timpanic surfaces (like the soundboard on a piano), and while adding mass to them helps, adding a mass with a vibration-attenuating adhesive like
Green Glue (tm) helps.
If you're re-framing a partition wall there are other schemes to take it down even further, (staggered studs, furring, partial-thickness batts, etc. etc.), but for retrofitting blowing the cavities with low to mid-density cellulose or fiberglass and air sealing is step-1, followed by double-layering the gypsum using flexible adhesives.