Cleanups are generally organized through a dive club, or something like a community college or public aquarium. Because they happen on public property there's usually a permit requirement, or at a minimum some sort of coordination with a municipal Parks & Recreation Dept. At least that's the case for most places in the USA, since beaches, marinas, public piers and other such places where a cleanup would be targeted are public property.
I like to help out with these as well ... I'll be doing one later today in fact. We'll be cleaning up a popular marina prior to them putting the docks back in the water in preparation for the upcoming boating season. To answer your question, we will have at least a few relatively inexperienced, OWD's along. We'll pair them up with more experienced divers, and they'll generally be there to act as dive buddies or "helpers", learning from the experienced folks how to carry and use goodie bags, lift bags, how to rig a lift (assuming we find stuff heavy enough to require a lift bag), and basically just how to use their skills in ways they haven't done before. It's a good way to help newer divers expand their skill sets.
Cleanups also need on-shore people to check divers in and out of the water, to help clean up the shoreline above the surface, someone to take pictures (you almost always want to document the work for public relations purposes ... particularly if you're working with a municipal P&R department), and arrangements to haul whatever you take out of the water off to the local landfill or transfer station.
To get the best answer to your question, check with your local dive clubs, or if you have a nearby aquarium stop by and ask them (they're almost always looking for volunteers), or inquire at your town hall ... they almost always have some department that cares for public beaches ... usually Parks & Recreation or some similar name.
It's a great way to have fun, learn some new skills, and do something nice for your community all at the same time ... if this stuff interests you and you plan to do more of it in the future, I'd also recommend taking your Rescue class as soon as it's practical, and possibly a specialty class in S & R ...
... Bob (Grateful Diver)