In some departments that invite visitors to give seminars on a routine basis, the seminars are organized by students, in some cases by faculty, and in some cases by staff. I have been a member of departments with all of these types of organizational possibilities, and I have been in departments that changed from one to the other (e.g. faculty used to organize the seminars, now students do; and vice versa).
In departments with student organizers, the philosophy behind this organizational scheme is that students will feel more involved in the seminars/visits and therefore will perhaps attend more of the seminars (without being coerced), and that this is a good thing because going to these talks is an important component of graduate education.
In departments with faculty or staff organizers, the philosophy is typically that this is a service activity and students shouldn't spend their time on this type of thing. I suppose there might be departments that don't trust their students with seminar-organizing, but I haven't encountered this myself.
I have heard both versions of "visitors are more like to accept invitations from students" and "visitors are more likely to accept invitations from faculty". I haven't seen that it makes that much of a difference, but it wouldn't surprise me if some people are more inclined to accept an invitation in one situation or the other. When I get an invitation, my main criterion for whether I accept or decline is "Do I have time for this?".
From my experience of having given > 60 invited talks at other universities and colleges in the past ~ 10 years, I can say that I see no difference in the quality of organization of the visit as a function of what type of academic creature does the organizing: students, faculty, or staff. There may be considerable variation depending on the individual's organizational skills, but this does not correlate with age or academic status.
The answer to the question about seminar organization will vary in part depending on the type and size of institution, but I decided to keep the poll simple. In places with regular seminar series:
7 months ago