When I was there, UT Austin had Sophex, a seminar organized and attended by first- and second-year grad students. I think it was advertised during orientation to let incoming first-years know about it. We'd meet at the end of the day on Friday and walk to the pub near the math building afterward. One of us would speak and another would bring snacks (as with Many Cheerful Facts). The atmosphere was friendly, and I always enjoyed going!

People would speak about things in their research area, things they found fun, or a mix of the two (as with U. Penn's Pizza Seminar). I think they generally succeeded at making their talks approachable for all the attendees, who came from a variety of fields. The past talk titles listed on the seminar page give a sense of what people spoke about.

Speakers could, and sometimes did, invite people other than first- and second-year students to attend their talks. I think they mostly did this when they were talking about things in their research areas, and wanted to share with coworkers in later years. I don't recall anyone inviting faculty when I was there, although I think it would've been fine. I think invited attendees generally participated considerately and tried not to dominate the conversation.

The organizer(s) would typically pass down their duties after one semester, e-mailing out a call for volunteers to replace them. As a result, we had a different organizer or pair of co-organizers each semester (although a co-organizer once became the next sole organizer). Both first- and second-years have organized; I've seen outgoing organizers ask specifically for first-years or rising second-years to replace them.

The seminar page seems to have last been updated in fall 2018, so I don't know whether Sophex is still running. If you've participated, feel free to comment with additions or corrections!