I live in a side by side. We can hear a lot of noise from our neighbours. We hear them walking around all the time. The worst problem is at night when we can hear them snoring through our bedroom wall. There is insulation in the walls already. Do you have any ideas for anything to help eliminate the noise at night? Cork on the walls? Carpet on the walls? Thicker drywall?
If moving to a single detached home is out of the question, you may be able to do a few things to minimize the noise from the other side of the house. I will attempt to suggest several methods that may help, somewhat.
Most side-by-side homes have a common wall between the two units that will provide a physical barrier, but may not provide adequate soundproofing. In many homes that I have inspected, this wall is constructed out of concrete block, which provides excellent security and fireguard but may have nothing in the way of insulation or soundproofing. Your statement that the walls are insulated may apply to the exterior walls of the home, but may not include this common wall. This wall is likely the area where much of the sound is transferring from the attached dwelling.
Sound waves will transfer though air, or a liquid such as water, but often will be loudest when going through a solid medium, due to vibration and other factors. The concrete block wall, likely in your home, will contain some hollow voids, but will mostly be a solid surface. This wall and the floors in the home will allow vibrations to be heard on both sides of the home and are responsible for the traffic noise you describe. Since the floor structures on both side of the home are attached to the central wall, a physical break to reduce vibrations will not be possible. The floor vibrations may be difficult to eliminate, but the noise through the wall may be cut down substantially.
Anything that can help absorb or reflect sound waves will provide some relief from the noisy neighbours. Your suggestions of adding carpet or cork to walls is thinking in the right direction. These may help, somewhat, but may not eliminate the vibrations through the wall. Any soundproofing that is installed should be held away from the common wall, to provide a physical break and reduce sound transfer through that medium.
Often, an additional insulated wall will be installed inside the common wall to help reduce noise. This wall should be built slightly away from the concrete block wall and have a foam gasket, or other physical break, installed around the perimeter framing to help prevent direct contact with the floor, ceiling and walls. The sheathing on this wall should incorporate some sound absorbing medium, such as the cork mentioned, for best performance. The drawbacks of building a wall like this are the cost and loss of living space.
If installation of sound barrier walls is not practical, use of sound dampening wall coverings alone may help reduce the annoying sounds. The suggestion of carpet on the walls sounds odd, but may provide substantial relief, if done correctly. Large throw rugs may be used if they are big enough to cover an entire wall in a room and should be suspended from the ceiling, slightly out from the wall. Burlap is often used for the same purpose or soft synthetic foam rubber. The difficulty with this arrangement is that most of these materials will absorb dust and dirt as easily as sound waves and will be difficult to clean.
Another suggestion to help with the noise, especially the nighttime snoring, is to rearrange your living space to minimize the affects of the noise. If the headboard of your bed is next to or up against the common wall the noises will be more noticeable. Moving beds away from the common wall and replacing with them desks, dressers or other large furniture will not stop the noises, but may make them less noticeable. Moving other furniture to strategically keep your family members as far away from the other dwelling, when relaxing, will also minimize the annoyance.
The last item to address is the relations with the neighbours. If you can hear their noises, chances are they can hear your family, as well. A friendly discussion about their attempting some of the simpler suggestion put forth here, also, will be a good idea. They may be oblivious to the problems you are encountering and may be able to help with some modifications of their own. If soundproofing measures are taken on both sides of the side-by-side home, they will be much more effective.