Do they discuss the graduate student in meetings with each other? Or do they have any obligations to interact with the graduate student in other ways?
I think the variations here are national rather than by discipline.
In the U.S., the thesis committee's primary role is that of a review panel when all is going well, and as a neutral arbiter between student and advisor when it isn't. Again, there is no formal obligation on the part of the committee—except when convened by the student or advisor, but this typically occurs on a more frequent basis (every one to two years or so). As individuals, however, committee members may be consulted with on various topics (job placement and career strategies, suggestions for future research directions, and so on); however, this is voluntary and, again, not very frequent—probably one or two times per year.
In Germany, by comparison, the primary obligation occurs at the time of the thesis defense, when they serve as evaluators. Outside of this, you might never meet with the professors on your committee, other than your primary advisor.
To complement aeismail's answer, I would add that in France, there is a committee that evaluates every year or so the progress of your PhD (but it's rather high-level, they are just looking for problems, like someone who would have stopped working), and this committee is chosen by the grad school.
The final PhD committee, who evaluates scientifically the PhD, is decided only at the final stage of the PhD, and it is proposed by the PhD advisor and then validated by the grad school.
To also complement the answers given, here's how it works at my institution:
There is a small committee of faculty members, chosen by the student but with particular rules established by the department (Three total, choose one member from List A, two faculty members, two must be tenured professors in the dept. etc) who determine whether you are ready or not to transition from coursework to working on your dissertation proposal, and the dissertation itself. This is often, but not always, a subset of committee #2.
The dissertation committee. This committee has several formal obligations toward the student:
- They meet with the student when they submit their dissertation proposal, to provide comments, criticism and ideas in order to strengthen the proposal, (theoretically) stop doomed projects from ever being started by the student, and identify any areas they believe the student may need additional coursework or expertise in order to complete their project. They also meet independently without the student during the same meeting to have a candid discussion of said student. This is usually about the project, but can range to cover mentorship concerns, thoughts on their job search, etc.
- The student is obligated to meet with each member of their committee every 6 months or so to keep them updated on their progress, and the committee as a whole meets at ~the midpoint of the dissertation project to make sure any problems are addressed before they have a chance to "go too far" and stall the defense itself.
- They then meet at the dissertation defense, evaluate the student, and make the determination as to whether or not they have successfully completed their program.
Certain members of the committee do have certain roles, beyond the usual advisor/chair role, but these are usually informal. For example, there is often a statistician on the committee, with the informal understanding that this is the person the student can turn to for questions re: statistical methods.