I'm in an MPP program and you definitely don't NEED a JD to do policy. That said, it depends on exactly what you want to do. If you want to do non-profit work, don't waste your money on law school. If you want to work on the hill, you could go either way.
I want to lobby and while a JD isn't necessary, it will help me get where I want to go faster (signaling, for you econ types). I also think a JD will give me a more complete perspective on policy making. I work in the health policy field, and have certainly noticed that most of the "important people" have JD's. I don't think that's just because it's the lobbyist pedigree; I think the perspective gives you an advantage to influencing the lawmaking process.
Also, in my experience, it's not that easy to get an LC job especially on the Senate side. There are tons of people vying for those jobs. Not only do you have to have the credentials and be good at what you do, you need to know the right people too.
One final thing- my biggest piece of advice to anyone who thinks they want to do policy work is: work in a policy/ political field first. (You may very well have experience in education policy already, in which case ignore this.) There is so much more to learn "on the ground" than can ever be taught in a classroom. And what is taught in the classroom is usually a good academic exercise, but not really how things actually operate. I'm not saying the MPP is a waste of time; in fact, I think it's an excellent degree to teach someone how to formulate and implement policy ideas in a more systematic, well thought out kind of way. You will definitely get more out of it if you can apply it to what you already know about the political process.
I have a lot of experience working with my state legislature and have been to DC three times to meet with our state's congressional delegation regarding education policy and from what I've seen almost everyone worth talking to has a MPP, MPA or JD. But before doing anything, you should consider an unpaid internship with your congressperson, they're relatively high-stress low-recognition, which is what you'll typically experience. Make sure you like that kind of work before sinking tens of thousands if not hundreds of thousands of dollars into graduate school.