There are a couple of explanations for your problem. The most likely is soot from either your heating system or a fireplace. There is (obviously) no insulation where the studs are and metal screws would be ideal for transmitting the colder temps from the unheated areas of your home. These areas will attract the humidity quicker than areas that have insulation. These moister areas will then attract the soot from your heating and from cooking.
There may be no way to stop this from happening, the only way to get rid of these stains is to repaint using a stain killing primer. The easiest stain killin primer I know of is Brunning's "Grabber Plus". If you use this stuff, be advised that it is also an adhesion primer. It really sticks to anything it's allowed to dry to, including and especially your skin!