Will adding batt insulation between the studs in my wall or ceiling block sound? I probably get this question at least once a day; the short answer is NO!
Lightweight porous materials such as fiberglass and open cell foam are great for reducing reverberation in a space by absorbing the sound reflections off of hard surfaces such as plaster, drywall or concrete block, but they will not prevent sound from passing through them. A material's ability to block sound is a direct function of its mass. Heavy dense materials will block sound; light porous materials will absorb sound.
So the solution is to reduce the airborne noise by adding weight to your wall or ceiling partition with a flexible mass loaded vinyl barrier, and reducing the structure borne noise by decoupling the partition with sound isolation clips.