In a for-profit context, worker cooperatives are firms that are owned and managed by workers. Pérotin (2012, 2015) summarizes research to show that worker cooperatives have positive impacts on both firm productivity and employee welfare; there is a lot more research showing that worker ownership is modestly better than regular capitalist ownership but I won't get into that here.
There are plausible reasons to explain why the private sector would generally refrain from adopting worker cooperatives even if worker cooperatives are better: owners and managers have a self-interest to keep capital and decision power to themselves, and people may have a pro-capitalist bias. So the rarity of this idea is not good evidence against its correctness. (Mild forms of employee ownership are fairly common, however.)
Nonprofits can't be owned, so a cooperative nonprofit would just be about worker management (i.e. democracy). This is called a Worker Self-Directed Nonprofit. I didn't find any research showing how well this idea works, though of course it seems good based on the results of for-profit worker cooperatives. If you search the phrase Worker Self-Directed Nonprofit there are various sources of (generally cheery) commentary and guidance. I'm not sure if it's been tried very much.
Workplace democracies, and WSDNs by extension, also seem to have a generally positive moral PR tint. Though a minority of people may perceive them as silly and inefficient.
Overall, experimenting with WSDN in the EA community seems like a valuable idea.
I haven't worked for an EA organization, I just wanted to throw this idea out there.