Is it legal for someone who doesn't write to sell their idea and have someone else write the book for them? Can they split royalty profits between each other?

Sure. You don’t think celebrities really write those books with their names on them, do you? The process is called “ghostwriting,” and it’s done all the time.

The sticky part is the royalty share, though—that all needs to be handled with a lawyer if you’re smart, but usually the way ghostwriting works is that the person with the idea pays the ghostwriter a flat fee to produce the book (a work for hire) and the finished product becomes the property of the person who paid for it. The ghostwriter has no claim on any future royalties generated by the book.

As far as just selling an idea and having somebody else write it, it can be done but most legitimate writers won’t go for it unless the money is very good. They have enough ideas of their own, so they usually don’t need to buy them from other people. But there’s nothing illegal about it, as long as everyone agrees to the terms.