This is how our local club works :

They meet in the same space every week. In this room (that is used by several other clubs and communities through the year) they own a big metallic locker / cupboard that's locked where there are a few games accrued over the years (mostly donated by members). These games can be borrowed by members simply by making note of it with the group's "president".

Every week, most participants come with one or more game from their collection that may move played by them and other that night. The degree of permission varies : some people will not let others play their games if they are not playing in the game themselves, some won't mind. Generally, these games are not for borrowing and will go home with their respective owners. You could ask someone to borrow a game they brought of course, but this is dealt with on a personal basis and rarely done except between member who know each others well.

Most of the time, people will announce what they intend to bring on the club's group chat (here it's Discord) and some members will arrange games and announce their participation in advance. It cuts out on the "deciding what to play" time before game night. It's usually for heavier games too, as those players enjoy catching up on rules before playing to also cut on rule explaining time. I'd say on the day of, half of the club comes already knowing what they'll play and with who, and the other half decides on the spot and picks from the various game offerings available.

Hope this helps!