If and when the plaintiffs receive a response to the subpoena, the plaintiffs will send a letter saying that, unless the person who owns the internet connection is willing to settle the plaintiffs will replace the name "John Doe" with that person's name. In cases which were initially filed in a court without personal jurisdiction, the plaintiffs will (theoretically) file a new case in the federal court with jurisdiction over the defendant with his or her name in the title of the case.
In these letters demanding money to settle the case, the settlement demands originally ranged from $300 to $1,500 per infringement. Recently, the primary plaintiff in Massachusetts has been Three Strikes, which targets IP addresses with multiple infringements. Their starting position is to demand the statutory minimum of $750 per infringement (knowing that they have probably have enough infringements alleged to exceed any reasonable resolution this way).
After that, the case will theoretically proceed as any other civil case would. The defenant must file an "answer" to the complaint, stating their position on whether the allegations are true or not. Then, once the parties are in contact with each other, each side can demand "discovery" from the other, whereby they can request that the opposing party be required to produce documents, answer questions and sit for "depositions." It is through this process that each side can develop their theory of liability or defense. In particular, in cases like these, it is in discovery that the plaintiffs can require the defendant to answer the questions 'did you in fact download this copyrighted work?' and 'if you did not, who else has access to your internet account?'