It's not too late to insulate.A few years ago, most American homes had no insulation in them. When a homeowner did insulate, it was usually with fiberglass batts.
Then along came some better products. One of them is blow-in cellulose insulation. Does it work? I can attest to this: YES. I've used it in walls and ceilings, and it works quite well for airborne noise. If you live in an uninsulated home and you hear your neighbors voices coming through your walls or ceilings, cellulose will dampen that noise or mute it entirely.
It doesn't work for all types of noise.But it doesn't eradicate impact noise. You might still hear the sounds of someone walking across a bare hardwood floor with their shoes on, for example, or hear wall-mounted cabinets opening and closing. There are other solutions for that, and by combining different solutions you can control most structural noise and create more privacy.
Low costBlow-in insulation is inexpensive. You can hire a specialist to do it, or as this YouTube video by Craig Wolfe shows, you can do it yourself with rented equipment. As you'll see in this video, cellulose provides much denser and comprehensive coverage than fiberglass batts. They actually blow additional cellulose into the wall cavity with a fiberglass batt already in it to show you how much air space was still available.
No demolition. Takes only a couple of hours.What's really great about blow-in cellulose is that you can keep your existing walls. You just need to cut some small holes into the drywall, which you can replace afterwards, and blow the insulation into the cavity.
This promo video on Cocoon Insulation shows how cellulose fiber is fire retardant and because of its density, very effective at noise control: