Every graduate student has the right to learn and the responsibility not to impinge on other students' right to learn. As a doctoral student, it's your responsibility to ensure that you don't get in the way of other students' learning by creating a poor learning environment, blocking access to information or otherwise acting as an obstacle to learning. This applies to other doctoral students and to other grad students and undergraduates.
As a doctoral student, you will be carrying out research during the bulk of your program. This needs to be high-quality research, because its quality affects the university's reputation. Your research needs to be fair, balanced, rigorous and objective. If your research reveals something that you object to, you still need to publish it; a major responsibility of a doctoral student is that he doesn't get in the way of his research.
While the university is a self-directed education system in general, doctoral students need to be even more self-directed than they were as master's and bachelor's students. You are responsible for setting your own milestones (after discussing them with your adviser) and meeting these milestones on your own. Nobody is watching over your shoulder and forcing you to work when you are a doctoral student -- you are responsible for being this person for yourself.
Many doctoral students also serve as teaching assistants. If you have this job, you need to perform it to the best of your ability by always being prepared for class, being receptive to students and giving constructive feedback whenever possible. As a doctoral student, you have the responsibility to deliver as good an education as you are receiving.