The supply pipes and waste pipes are entirely separate systems.
All of the waste pipes have to come together eventually, since you have only one connection to the sewer. Usually, this is done at convenient places in the design, and using connections that minimize backing up into other drains. Introduce a clog, lets say 1ft past a connection, and pump in a bunch of pressure, all the stuff behind the clog will go up the line to the other drain. Stupid, stupid way to get rid of a clog, but fast, which is good for the lazy super.
I once had a clog past a joint, it resulted in kitchen water backing up into my bathtub. The sink and tub shared a wall, so their linkup was pretty close to the drains.