There are a couple of products you should research, Green Glue and QuietRock. Blown-in or batt insulation doesn't substantially increase the mass of the wall or interfere with the path of the noise, so I don't think it will do much to reduce the noise level. Increasing wall mass and acoustic decoupling are two ways I've seen to effectively sound-proof, which are how the products above function.
Adding insulation to interior walls shouldn't cause problems if the HVAC system is properly designed and keeps the rooms at similar temperatures. Check that the empty wall cavities aren't being used as air-returns to your HVAC. If they are you will need to provide a clear path for the air return if you insulate.
Just to add to the above, which is good info, watch for sneaky paths of noise to travel in, like electrical outlets. Apparently FG will cut down on some of the lower frequency sound waves. This was just discussed on greenbuildingadvisor.com, and one of the cats there mentioned that about FG. In general, as mentioned above, add mass.