Their job would be to engage with the public. This would be nothing like a Public Relations guy giving a statement, customer service giving a run around, or a press release. They would spend their time reading comments by the public -- both in public places and sent directly to the company -- and having conversations with those people. They would not give a statement and move on, instead they would actually engage with what the person was saying.
In some circumstances, this would be a far more effective use of money than advertising. There are all these people who want to interact with the company. So why not hire people to tell them a personalised version of the company's point of view?
A good arguer would persuade a few people that the company was right, and a fair amount would become less hostile. But more than winning arguing points, he'd show the company *has* arguing points that can hold up in a sustained debate and don't fall down after a few back and forths. He'd be demonstrating that people seriously believe the company is right and have thought it out.
Another part of his job would be to relay any opposition to the company that he considered especially interesting or thought had a good point. A company has a hard time reading and filtering a huge in basket, but when you cut down incoming arguments by a factor of a thousand or so (removing duplicates and bad arguments and fluff) it gets way more manageable.
Unfortunately the primary problem I see with this idea is the difficulty of hiring qualified, competent arguers. Letting people speak for your company is risky, so you need to be sure they are good at it. And the arguers job requires a lot more skill to avoid mishaps than a press release writer's job. The arguer will write a thousand times as many words, but every single one could end up quoted by the press if he messes up.
One day in the future, arguers will not be expected to be perfect and if they messed up now and then the press would realise this doesn't reflect badly on the company.