I have an arrangement with a guy that's sort of like this. He's awesome at sales, and he had a company that did a lot of email marketing for brands - so I started talking to him about offering other services to those clients and he made a new company for that side of things. He handles sales and payments, pushing everything through the smaller company he made so he can more easily get away with charging agency prices rather than freelancer rates. In reality, it's him and his secretary/project manager plus me and my outsourced help.
He takes around 20-40% of most projects (depending on size, margins, and price sensitivity of the client) and handles sales, collections, and payment processing. It worked really well for about 5 years until he had some weird issues with a bad employee at his main company, but that's totally unrelated to the validity of the arrangement. Now I'm on the fence about it.
The only big issue I've had to deal with is that it's really hard to teach someone what they need to say or not say to close a deal. He has occasionally given clients the wrong idea about things, over/undersold potential results, or misread the scope and needs of certain clients, and that can make things harder on me. Overall, though, it's been a nice way to get more business than I would have otherwise gotten and I don't have to rely on some low-level salesperson who reads from a script. I used to do all my own sales, but I don't particularly enjoy it and this lets me devote a lot more time to side projects while still getting lots of household name clients. I like just giving him a quote and then hearing back in a week or two, "Ok, we closed ---, go for it." Occasionally I'll join in on a call or meeting if the client needs more persuading, and I like that our arrangement has that flexibility.
I will say that I don't think this setup is terribly common, though. A lot of people get regular work from agencies, but it's usually a different kind of thing where they work for a small hourly rate (rather than a partnership with the owner where you take the lion's share of the fees charged to the client and they sell what you tell them you can offer). I think you'd have to either train your own salesperson as an employee/contractor, or find someone who's connected and has a steady base of clients but who doesn't yet do what you're offering.