Definitely depends on the size of the school. I went to college to teach high school drama, and most of the schools that hire actual drama teachers to teach classes full-time are either very large or very well-off. Many schools will have one or two drama classes but nothing very extensive. And many other schools have no graded drama programs at all. In that case, a drama club can be supervised by basically anyone who volunteers to step up. (One high school where I grew up had its shows co-directed by a science teacher a history teacher.)

Either way, samisbond's comment hits the nail on the head that "elective" is not the same as "extracurricular." When drama is offered as a class, it is nearly always going to be an elective because no school is going to require all its students to take it. (There might be exceptions but I don't know of any.) However, they may need to take some sort of humanities or artistic expression class and can choose from several options, including drama.

Teachers typically direct the shows just because a lot of schools don't have students ready to direct their own shows, but there's nothing that would forbid that in most cases. The school where I student taught in college had major productions directed by the drama teacher and then several smaller productions directed by drama club seniors. I've seldom run into a full stage production directed by a student, but a school with a more serious arts program or a lot of talented students might go for it.

Props/costumes/tech vary greatly from school to school. In terms of design and prep, that is usually either done by students or volunteers (students' parents can be a great resource for this). Costumes are often rented or bought inexpensively and used for many shows if possible. :-) Not a lot of schools can afford to hire someone for a tech show, though some larger or arts-centered schools may have a theater tech teacher who has their own classes and steps in to do a lot of it. In terms of running the show, like dressers or lighting changes, that's typically done by students.

So there's a lot of variation depending on the size and financial situation school, the support from the community, and the students involved. I'm not teaching drama correctly but over the past few years I've spent a lot of time working on productions in a couple different high schools, so feel free to PM me if you have any more questions.