I think there are 2 main reason that it is harder for international students to get in schools in the US:
1) people here probably don't know your recommenders unless they are famous OR you are applying to the experiment your recommender is working on (in the case of big collaborations).
2) the expenses that the university covers are higher for international students than for domestic, i.e. you are applying for a quite a bit higher paying job than a US student is when applying to grad schools here.
Now, since private schools are private
they don't really depend on the state or gov't to fund them that much so they usually accept more international students than public school. Because of that you are more likely to get accepted in private school than a public if they both have the same rank otherwise...but you already knew that... so just pick some private schools.
As for making the choice, you might check out the World News Report on who is the best on String theory and just pick some of the schools down the list as safety OR you can wait for the NRC report to come out in November hopefully (however, it is not separated in subfields).
Also I honestly think the best way is to actually look onto school's web pages and talk to prof's who are in your field. Ask them what their research is about more specifically and also if they have room for more students this year. The tricky part is that some schools accept people on general basis (i.e. they will choose these with best profile and let them in) while others will choose on the basis of whether they have spaces open for someone with your interests.
So keep researching. Ask around. Maybe ask professors at your schools that are involved in the field you are interested. They will know who is doing what. Physicists do gossip!
Finally, take this with a grain of salt... I am no expert