Conceptually, if you don't care about speed, take an algorithm that decomposes $N$-gons into $N-2$ "disjoint" (modulo common edges) triangles -- for convex polygons this is trivial -- and then, for each pair of triangles (one from $P1$ and one from $P2$), apply an algorithm that computes the overlap, if any, of two triangles. This shows the complexity is at worst $O(mn)$, at least in the convex case. One question might be, how does this compare with the approaches in the answers already given by meij and Stephen Sturgeon?

Conceptually, if you don't care about speed, take an algorithm that decomposes $N$-gons into $N-2$ "disjoint" (modulo common edges) triangles -- for convex polygons this is trivial -- and then, for each pair of triangles (one from $P1$ and one from $P2$), apply an algorithm that computes the overlap, if any, of two triangles. This shows the complexity is at worst $O(mn)$, at least in the convex case. One question might be, how does this compare with the approaches in the answers already given by meij and Stephen Sturgeon?

Added later: It occurs to me I might be under a misimpression as to how the polygons are being specified. I am assuming that to "have the boundaries" means, in effect, that you have a listing of the vertices as you go around the boundary. If it means something else, then what I described above requires a prefatory step that produces such a list. (Also, if the OP really wants the overlaps of the convex hulls of $P1$ and $P2$, then lists of the hulls' vertices need to be ascertained.)