Princeton has a Graduate Student Seminar which, at least when I was there, was a lunch seminar offering pizza. It is part of the pure math department but I think people from the program for applied and computational mathematics sometimes attended or spoke.

I attended a seminar that was designed for a mix of mathematicians and physicists (not grad students, mainly postdocs), which seems similar to your idea - based on this I have some thoughts.

(1) Whether such a seminar would work at your university depends a lot on the culture and structure particular departments you want to include. The more interaction there already is between pure and applied math people, the easier it will be. At a minimum you need the pure mathematicians to be interested in learning some applied math and vice versa. Mentioning statistics and string theory suggests you may be interested in more departments than just those two, although maybe applied math at your university includes those.

(2) I think, if you want a seminar uniting two groups, it is vitally important that you make clear to the speaker and the audience that talks should be pitched at whatever group the speaker is not in. So pure mathematicians giving talks should focus on what applied mathematicians can understand and not try to necessarily say anything novel to other pure mathematicians. To help with this, you'll want to try to get people from each side asking questions about the other. The organizers of the seminar can start this off by asking questions of their own (but probably not asking questions on talks whose material they know well).