For much of my career, this seemed like a silly question. Of course grad students should publish journal articles, book chapters, etc.. I was socialized to believe this. My first few publications originated in term papers that my professors insisted I submit to a journal. I think I had four journal articles out before I applied for my first teaching position.
When teaching at my prior university, I was surprised to learn that some of my colleagues discouraged their students from publishing papers. Grad school is to read and to learn and to write a dissertation. Working up papers for publication takes time away from the important tasks, so students should not be encouraged to waste precious time on publications. Once the Ph.D. was in hand, students can publish all they want (and presumably they would spend less time at that point hassling their professors for help in turning term papers into articles). Needless to say I disagreed strongly with this view of student publishing. If grad students don't publish anything, how will they stand a chance in today's job market?
When I moved to ASU, I was pleased to find an academic culture that encouraged students to publish, and I found the extent of student publishing (at least in archaeology) quite impressive. The question of student publishing did not seem to be a concern anymore. Of course students should publish. But then at a party last week, two younger colleagues said they had been discouraged from publishing while in grad school. These were major departments at top universities. Publishing takes up precious student time, and students aren't smart or accomplished enough to publish anything interesting. I was flabbergasted. My view is that if a student's term papers aren't good enough to turn into journal articles with a little work, then they are probably wasting their time with graduate study.
So, for you students out there: Get off your rear and start publishing. And let me modify my second sentence above: you should definitely publish journal articles, but think twice about chapters in edited volumes. The latter will take longer to appear in print, they have less prestige, and fewer people will read them. See some of my earlier posts on edited volumes, Here, or Here.