Let me to qualify at least my claims, dear friends.
1. Of course, there are fewer international applicants than American ones. However, there ARE reasons why foreigners without permanent residency/citizenship are indeed disadvantaged in applying to schools below the top few. One of them being that most schools make NO effort whatsoever to help such students pay for law school and are aware they can't afford, which is why they just don't admit them. Most law schools past, like, top 10 don't even mention international students on their websites!
2. That said, I personally spoke with the HSL associate dean of admissions in the fall and asked her why there are still so few internationals. She, of course, didn't give a straightforward answer and said something about law school being expensive -- and this is a school that actually does make the best effort to make legal education affordable, even to foreigners (i.e. co-signs loans, etc).
3. In conclusion, yes, Yale and Harvard and Columbia like bragging about their diversity and how many internationals they have. Their undergraduate schools do, too. As soon as you go past the top 10 schools, however, or even top 6, this is not at all the case. If you have amazing numbers, then yes, you can apply and probably get into a T6 school, which will admit you irrespective of nationality and co-sign your loans. If your numbers are more in the pretty good but not amazing range, you lose competitive advantage over Americans to lower-ranked schools.
This is my understanding and you can try to argue, but I've been there and done that with applying to colleges four years ago. I remember recently reading what George Washington Law says about internationals -- it goes along those lines: "If you really wanna come here, and like, have really good numbers, sure, like, we'll take you, but you have to, like, figure out a way to get $200 000. K, cool, anticipating your application!"