In the U.S. it really depends what state you're in. Some states have co-ops as a specific business vehicle and in many of them the co-op model is very different. There are big farm co-ops, EMCs, etc. in the U.S. (Land-o-Lakes and Florida Natural Oranges come to mind, I believe they're both fully farmer owned), but I'm not super familiar with them or how their model works, but in Georgia where I am at least co-ops tend to work in one of two ways:

For smaller co-ops that don't bring in members very often or who are just getting started they often create an LLC and put in the articles of incorporation that all future members will receive an equal share and equal voting rights. This does require filing paperwork for every new member, but keeps taxes easy.

For co-ops that expect to have lots of members rapidly (consumer co-ops that sell shares in their store to whomever wants one, for example) they normally form a corporation and allocate a certain number of shares up front and indicate that any member has equal voting rights, etc. this makes the taxes harder, but is a lot less paperwork since you have to file stuff about the number of shares sold at the end of the quarter (or whenever, I forget the exact reporting details, been a while since I've done this) but not every single time you add a member.

Georgia (and a lot of other states) also have specific laws for EMCs and other specific types of co-ops, so it's really fragmented. Take this with a grain of salt, it's just two examples of how a lot of smaller co-ops do things.