I've had a similar problem -- not co-working like one big open room, but our group rents individual office rooms from one of those places that provides rooms, furniture, and administrative support, and the build-out of the space was obviously done by the lowest bidder. The walls are thin, the speaker-phones are loud, and noise carries.
There's one particular guy near us who tends to have loud phone conversations. The speaker is turned all the way up, he shouts into the phone, and he leaves his door open.
Early in our time in the space, when one of his meetings was preventing me from being able to get work done several rooms away, I walked over to his office, made eye contact without saying anything (didn't want to interrupt), and closed his door. I naively assumed that this would be sufficient, but it wasn't. A few days later the same thing happened and I again closed his door. A few days after that, our local lead told me that the landlord had received a complaint from him about me, so we decided that trying to engage with him directly any more would not work.
So we've relied on the landlord's office manager/administrator since then. When he is being disruptively loud, one of us talks with the office manager, who quiets the noise. She's reportedly had several conversations with him about this behavior, and the frequency has been going down.
The problem, I think, with my directly approaching him is that we, like you and your noisy neighbor, have no relationship, no shared employer, and no accountability to each other. Your loud guy (and mine) are probably operating on the assumption that they're paying for a working space, working involves phone calls and meetings, and that's normal and justified. The office manager's job in a place like this or co-working includes dealing with issues between or among tenants. So I recommend going to that person, not the noisy neighbor directly.
How you bring it up with the space manager is likely to have an effect on your results. Don't storm up to the person and say "that jerk is yapping on his phone and I can't work; do something about him". Instead, calmly say something like: "I'm having trouble focusing on my work because of noise from one particular source. I know co-working spaces are noisy, but this person's meetings are more than the norm I'm used to. Could we find a solution to this problem?"
You have a solution in mind (send the guy to a meeting room), but I've found that it works better to talk about the problem* rather than a proposed solution. Give the manager a chance to understand the issue before you jump to your preferred answer. Further, if the manager proposes the obvious solution (which you'll of course agree to), then the manager feels in control and you have grounds to say "can you talk with him about that?".
You want a good, cooperative relationship with the person who controls your space. I have a good relationship with mine, and I think taking this approach to problems in the space has a lot to do with it.