A lot of the beekeepers here paint their boxes inside and out, generally with acrylic paint. Before I get flamed, most of the store bought boxes are made from radiata pine and then treated with copper naphthenate wood preserver to make them last a bit longer. Radiata doesn't stand up to the outside environment very well at all without some treatment and I suppose the paint inside the boxes protects the bees from any ill effects of the copper naphthenate. The acrylic (water based paint) doesn't worry them, but the boxes stick together until the paint is well cured. This may take several weeks outside in the sun and rain and you have to keep the boxes apart while this happens or your paint job will be marred. Like AllenF said, some missing paint from where the boxes touch isn't normally a problem. If you are a bit impatient, after the paint has dried for a couple of days, you can smear a very light coat of vaseline/petroleum jelly around the top and bottom of the boxes where they touch and then they wont stick together. Just be careful to use very, very little as the vaseline will make the boxes slip apart and your nice neat stack/hive will fall over really easy. It will prevent the bee's propolis from sticking too.
On the primer point, some of the exterior acrylic paint is self priming (eg. Wattle Solagard here) and it sticks pretty well once it's cured. I have painted a timber dinghy with it and the rocks have a hard time removing any paint from the bottom. This type of paint is pretty thick (too thick generally) and so covers pretty well without any primer undercoat.