but which medical specialty would consentrate on STD's? (there is now a specialty for public health/communicably diseases, isn't there?)

Usually it's referred to an ID person these days (Infectious Diseases). But - at the risk of sounding like Dr Bill - part of my time in the Army was spent as I Corps area dermatologist, and indeed VD patients came to my clinic. Although there wasn't much siphilis, mostly the clap (that's gonorrhea for you innocents). Anyhow the treatment of the one covered the other as well so it didn't matter that much...