That is kind of what I was thinking. Would have to be something kind of simple with just a few parts of roughly similar material value.
Something like one of these tool makers clamps would be a possibility. It could be done a few ways probably generating 5-7 pieces with some wiggle room in division of labor. Something like this also favors using different machines, so someone who only has a mill or just a lathe can still participate, and possibly gets a tool that would be difficult to make in their shop. Also small enough that a guy with a Bridgeport and a guy with a mini-mill could both participate.
Lets call it 5 pieces, so you make 5 of your parts, and everybody gets what they need to build a clamp, make 10 and everybody gets a pair of clamps. Five is small enough to be able to find enough interested parties, but also small enough that if there were a lot of interest it could be broken into multiple project groups (2 groups of 5, 3 groups of 5 etc). Cheap and simple enough that if a participant fails to come through with their part not really a big deal, someone else can step in and make the parts or it can just be finished on your own.
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The clamps are just to illustrate the kind of thing I think would be a good starting point for such a project. A fairly simple project to help smooth over the inevitable bumps. If it works then it could get more ambitious, if it fails no one is out much but some time and material. The actual item built could also take advantage of what individuals bring to it, machines available, skills, comfort levels, desired objectives.
I think one of the major learning benefits of group project like this would be having to build to a spec. No fudging and hand fitting, because you don't have the other parts until, everybody is done.