For a concrete example, you might want to check the Non-Adult Sandbox. (Yes, the Non-Adult RPs are available to unapproved members.)
In general, Person A posts a request thread, either in One on Ones Wanted, Group RPs Wanted, or the Non-Adult Requests.
Interested parties respond either by post or PM, and negotiate from there. In the case of Adult RPs, a thread is then set up in the Adult side of the forum (not visible to Unapproved members or guests), where the RP itself takes place. In the case of Non-Adult RPs, the thread is set up in the Non-Adult Roleplays board (just below the request board).
Co-writers alternate posting, back and forth, or (in the case of a group) sometimes arrange a 'posting order', detailing what their characters do or say, and what is happening around them.