By comparing the relative and Kuznetsov trace formulas in the global setting, Jacquet developed a method for characterizing the image of the base change map associating automorphic representations of $U(2, {\bf A}_E/{\bf A}_F)$ to automorphic representations of $GL(2,{\bf A}_E)$. In this talk I will discuss recent work of mine defining, proving and comparing local versions of the relative and Kuznetsov trace formulas on GL(2). When evaluated with matching functions, the local Kuznetsov trace formula and the local relative trace formula are equal and thus there is an equality between their local distributions on the spectral sides.