K-2 concentrates in the saliva in order to mineralize calcium in our teeth. Zinc supplementation increases salivary calcium levels: http://jdr.sagepub.com/content/84/1/35.short
Weston Price treated caries with his high vitamin butter oil, which contained a lot of k-2, but I wonder if correcting a zinc deficiency at the same time would make this process faster/more effective. The increased salivary calcium is likely for the proper functioning of taste buds, but I'm sure it's in a form that would mineralize. Most people coming from the SAD are going to be deficient in both of these (k-2 because most avoid a lot of egg yolks, butter and organ meats and zinc because of the high phytate content of the SAD, reduced red meat intake and also organ avoidance).
For what it's worth, zinc picolinate appears to be the most bioavailable form. I have encountered no stomach aches as a result of taking it without food.
Edit: I should also mention that zinc should be taken away from the copper-rich foods in your diet. The most concentrated source of copper that we eat is probably liver, which itself has a lot of zinc as it is, so the "advertised" levels of both in liver are likely overstating the bioavailability of each. If you took zinc at a time when you eat liver, you might negate most of the copper.