Here is the naive way (hoping for a more idiomatic, functional soln from someone).
p1 = {{1, 2, 6}, {1, 2, 4, 6}, {1, 2, 3, 5, 6}}; p2 = {{3, 5, 6}}; list = {} Do[ AppendTo[list, {}] Do[ If[IntersectingQ[p1[[x]], p2[[y]]], Intersection[p1[[x]], p2[[y]]] // AppendTo[listAppendTo[list[[y]], #] &, Nothing] , {x, 1, Length[p1]}] , {y, 1, Length[p2]}] list ```