What percent of graduate schools admit students at the department level?

[InquilineKea]

And what is the best way to inquire after whether they admit at the departmental level or not?

E.g.

At Princeton, the major part of the admissions process happens at the departmental level. The department reviews all applications, and then recommends that the graduate school offer admission to the very best students (regardless of the particular professor they want to work with).

I've heard that in other schools, then if a professor REALLY wants a student, they can get the student (I believe this is the case for the University of Washington Atmospheric Science department, as an example).

[Eigen]

All schools that I've heard of admit at both the department level (initial review) followed by a graduate school approval process.

This is not mutually exclusive with the second part of your question. If a professor really wants someone, they put pressure on the department's admission committee to admit them.

[DeeLovely79]

It all depends. If the program that you are applying to is a school-wide program then there may be an admission committee for the program that makes admissions decisions. The prospective applicants are then sent to the department they would enter for final approval. The only way to find out who makes these decisions is to ask admissions staff for the program.