Does someone even need a JD to be a law librarian?
Okay, this is a piece of my anonymity I wasn't planning to give up anytime soon, but in the spirit of helping random strangers on the internet: I'm a librarian and spent a couple of years on short-term contracts with a whole bunch of different government agencies and university law libraries, though I'm not doing that right now. (90% of librarians are female; the 10% of male law librarians mostly don't look like Fraggles; so if you're thinking right now, "Hey, that Fraggle law librarian dude who helped me with ____ might have been the Yogurt Baron himself! No. No, it can't have been. What are the odds?", the odds are actually incredibly high. It was either me or, like, one of four other guys, all of whom I know. We get together and talk about being male Fraggle law librarians.) So this is something I actually know a bit about.
For most law librarian positions---or even positions in libraries in fields like copyright---a law degree is a very strongly preferred qualification. If they can get someone with a law degree, they'll take them, period. That said, there aren't that many librarians out there with law degrees (and of the relatively few librarians who have law degrees, many got their MLIS degrees after quitting the law to pursue a completely new career path, and don't want to be law librarians), so oftentimes, law libraries will take whatever schmuck they can get. (Evidence: me.)
If someone comes out of a JD/MLIS program wanting to pursue librarianship rather than law, and they're genuinely excited about law librarianship, they will have their pick of any entry-level law librarian position, and eventually, they will have their pick of any law librarian position, because there are so few people with those qualifications who choose that path. Even if they graduate at the bottom of their class and hate the law and hate libraries and are a miserable dolt and/or oaf...they'll still probably be set, because there's so little competition out there.
Whenever law librarianship comes up - here or anywhere - the first question anyone asks is, "Why would you want to make seventy thousand dollars a year as a law librarian when we all know that getting a law degree is a 100% guarantee that you will make seventy trillion dollars a year on Bay Street?" But one thing we've seen a lot around here since we started getting more lawyers is a gentle deflation w/r/t reasonable salary expectations. Law librarians (generally) enjoy a pretty sedate pace*, a strong work-life balance**, and, relative to how much money actual people make in the actual world, very good pay. Sometimes, for very senior positions, we're talking about very good pay. There are occasions when a school or the government absolutely needs someone with both the JD and the MLIS to fill a position, and they're willing to shell out what they have to in order to attract anyone with the qualifications - not "Suits" money, but certainly "my blue-collar dad would consider you to be not just a middle-class professional but an actual rich person" territory.
Now that I've outed myself, I'm happy to take any questions about the career path of law librarianship, though if they're anything like the questions I took when I was a law librarian, the answers will almost certainly be, "Um! That's a good question! Which I certainly could answer. But I think you'll find the answer within yourself. Or Julie. Go ask Julie. Julie's great." (I'm a total pro on knife law, though. It's...it's too bad that thread got closed. Before I could answer the question about knife law. Which I could have done.)
*I have anecdotes, but they're defamatory of the other Fraggles.
**Oh! Oh! Here's an anecdote that's defamatory of me. I once spent an entire day building a secret path via which I could sneak out of the library a few hours early without my boss detecting me. (It's called "flex time".) Then, lo and behold, it's 3:15, I'm sneaking out on my path, feeling like a genius, and who happens to be on my path? My boss. Who is giving a tour of the library to some dignitaries. "Over here, we have some law books. Over there, some other law books. This appears to be one of our law librarians trying to sneak out of work. Nice fake mustache, by the way." The amount of trouble I got in was...astonishingly minimal. It's probably different in firms---I would guarantee that it is different in firms---but in most government/special law libraries, a lawyer needs something urgent right the fuck now 3% of the time, and the other 97% of the time, you're taking lunch whenever you're hungry and noodling around on whatever project isn't due for three months and hoping the 3% happens when you're on vacation.