"How do schools treat foreign students?" The answer to this question is another question: "How diverse are you?"
Ten years ago, Eastern European immigrants applying to law school were very rare; now they're common. The same is true of Chinese, Thai and Vietnamese. South American applicants have been common for a long time. "Melting pot" immigrants -- western Europeans -- are uncommon nowadays. But who knows what next year will bring?
If you're a recent immigrant of non-white ancestry, be sure to read what I've said about race, too.
Virtually all immigrants want to attend big name schools on both coasts. That's where their communities are, and those are the names their families in the old country know. Therefore, these schools have their pick of foreign students, as they do with every other kind of applicant. Lower LSAT scores are rarely forgiven. In addition, there's a decided disadvantage if you attended college in a foreign country, since the admissions officers know much less about the University of Krasnoyarsk than they do about Grinnell, or even Abilene Christian.
Moreover, altogether too many recent immigrants have the same story to tell: "I got here on a scholarship to study science and want to be a patent lawyer." "In the old country, my father was persecuted, we often went hungry." I'm not belittling those answers, or the realities they represent. I'm merely pointing out that they're, well, generic. Find something more to say, or your numbers will control your acceptances.
If You Weren't Born in the U.S.
Residents of their country of origin who are seeking admission here as foreign students, as well as recent immigrants who are U.S. citizens, are treated somewhat differently from people born here.
Make sure your application indicates when you learned English, and whether you spoke English or another language at home. If your parents did not learn English as quickly or as well as you did (as is often the case), you may have had unusual responsibilities as a child. Many ten year old immigrants do the banking, shopping, and paperwork for insurance and auto licenses for their entire family. If this was true for you, make sure you include it in your personal statement. If your background offered other unusual circumstances, make sure you let the admissions officer know.
Citizen, visitor, or permanent resident?
Differences in legal status generally don't seem to matter much (as long as you're here legally). That may affect your financial aid package, but not your level of diversity. In fact, when there are a lot of applicants from a particular country, those minor differences become irrelevant. In the last two years, I've had Eastern European clients who had F-1 visas, green cards, and citizenship. If I've seen all three, the law schools certainly have.
2005 NOTE: We're at war. We may not be at war with your country, but it may not matter. It is becoming so difficult for foreign students to get visas that some law schools are declining to admit them; they cause empty spaces in the class at the last minute, when school is starting and the visa hasn't been issued. Before applying to any law school, I would suggest you try to get the admissions professional to give you an honest answer about policies regarding foreign students.
2010 Note: If this situation has changed, I haven't heard it yet. More of my clients who are sure they're getting a green card find out they're wrong.