If such successful seminars exist, I would love to hear the details how this works in practice; what do students speak about, their research or something else; are such seminars run by students themselves or by staff.

The Math Department of the University of Colorado at Boulder has a seminar like this, run by graduate students. To answer the question, ``What do students speak about'', let me copy a few of the recent titles (= Fall 2021).

Wed, Sep. 1 5pm (MATH 350) Levi Lorzenzo (CU Boulder)
Connes' Embedding Problem, Kirchberg's conjecture, and MIP*= RE

Thu, Sep. 16 5pm (MATH 350) Peter Rock (CU Boulder)
Gauss's Greatest Hit (or Another look at Gauss's Theorema Egregium)

Wed, Sep. 22 5pm (MATH 350) Andrew Campbell (CU Boulder)
Party Hardy Spaces: An Introduction to Complex and Real Analysis

Wed, Oct. 6 5pm (MATH 350) Chase Meadors (CU Boulder)
The transfinite subway to Hilbert's Hotel

Wed, Oct. 13 5pm (MATH 350) Emily Montelius (CU Boulder)
Fault Free Tileability of Rectangles, Cylinders, Tori, and Möbius Strips with Dominoes