The bipartite distinct distance problem asks for the minimum number of distances determined by two finite sets in the plane. There have been several results in the case where one or both of the sets satisfies some algebraic constraint. In this talk I will give an overview of the problem, and discuss my result (joint with Brian McDonald, Jonathan Passant, and Anurag Sahay) in the case where one set is contained in a circle and the other is essentially arbitrary.