I gave a long post at the top level about running a book-focused reading group at a company, but your group sounds more like a Papers We Love[1] chapter.
I used to co-host the San Diego chapter of Papers We Love[2], and here's my secret sauce: I offered to meet with every presenter in advance for a dry run of their presentation. Probably two thirds of the presenters took me up on the offer.
For the group and the presenter, going through a dry run had the positive impact you would expect on presentation quality.
The benefit for me was that I got one-on-one discussion/learning with a wide variety of people passionate about a broad range of papers, and I also got to go through the material twice. So I learned much more and retained it better.
[2] https://www.youtube.com/channel/UCYPNnoVAQqb2Mue9v9f2euA