All surfaces absorb sound to some degree, and sound waves go through solid objects just like they do through open air, although solid surfaces absorb more of the sound. Unless they have significant insulation or consist of solid rock or masonry, most walls allow sound through to the other side.
Sound travels in waves that change as they meet different surfaces. The more pores a surface has, the more sound that surface absorbs; the thicker a surface is, the more sound that surface reflects. Many walls feature a thin veneer with a hollow interior, so while some echoing occurs, much of the sound continues through into the cavity inside the wall, and then through the veneer on the other side.
Adding insulation to the inside of a wall reduces the amount of sound that passes through it. All sound waves lose power over time, and insulation and other porous materials absorb sound. The waves enter the pores and ricochet, gradually losing power until they become inaudible.
Another way to cut the sound that comes through a wall involves separating the link between a ceiling and wall and adding an adhesive that creates a gap between the two before installing new drywall. Sound insulation clips and other devices are available as well, as they contain materials that absorb sound and keep it from traveling through walls.