I would add one comment to this, which is that I'd bet the number you saw (1,000) only counts individuals who made it to the first stage and were accepted by their universities as endorsed applicants. For some large schools, you have as many as 40 people competing for that endorsement. Those numbers would be impossible to count, exactly, because those individuals competing for the endorsement never actually submit on the official website, only through their own universities' processes.
From the Rhodes Press Release for this past year: "This year approximately 1750 students sought their institution’s endorsement; 857 were endorsed by 327 different colleges and universities." (http://s3.amazonaws.com/rhodesscholars-fileshare/final_press_release_2013.pdf)
So, statistically speaking, more realistically you're looking (at most) at a 1-2% rate.
As far as the regions go... They design the regions to be relatively equal in terms of numbers, so statistically your chances would be the same in every region. That being said, there's always been the question of whether a region that includes a state like New York is more competitive than (for example) one of the midwest regions (just because of the quality of the schools in each area).
Hope this helps!