In second draft, we switched off. In subsequent drafts, we did it all together at the computer, except for individual sections that one or another volunteered to take away and work on according to agreed-upon criteria and aims. A common intention is paramount. You have to be intending to write the same novel and not each lobbying for a different novel. A great deal of serious and honest conversation beforehand and a fair amount of compromising even before you begin is absolutely essential.
People look to the novel for a singular vision.
But if you're passionate about the project, proceed. You always have the option, later on, to invent a pseudonym. You say you gave this persona a name and a biography. I do pay attention to the rhythms of prose, but not in the same primarily oral sense as poetry. Mostly we talked about the work when walking or sitting at the computer.