Mathematica's region / Boolean CSG stuff is sadly very buggy, even in some simple cases like this where you really wouldn't expect it. I'm hoping it improves in future versions. To work around this I discretize the mesh into polygons and intersect each polygon individually, building up a list of EmptyRegion[3] and lines. The empty regions are discarded.
mesh1 = DiscretizeRegion@pr1;
prims = MeshPrimitives[mesh1, 2];
intersections = DeleteCases[RegionIntersection[#, pr2] & /@ prims, EmptyRegion[_]];
Graphics3D[{{Red, EdgeForm[None], mesh1}, Blue, pr2, Yellow, Thick,
intersections}, BoxRatios -> 1]
