Obviously, most of what follows is specific to france.
In common-law countries, civil liability follows from a specific legal theory. One has to state one’s legal theory (for instance: breach of contract), and then prove each of the various elements of that theory.
In France, and I believe in other civil-law countries, civil liability follows a more general pattern. Civil liability is established by a three-prong test of (1) wrongful action, (2) damage, and (3) causation link between the wrongful action and the damage.
Now, in most civil cases (>90%), proving prong (1) will rely on certain civil statutes, in which case the distinction is mostly theoretical, because those statutes more or less mirror common-law causes of action. However, a criminal act would satisfy prong (1). Civil law does not see much difference between damages resulting from a "civil" cause (e.g. I bought an item from BigCorp that turned out to be defective and I had to buy a new one) or from a "criminal" cause (e.g. BigCorp intentionally sold me an item that violates the electrical code and it set fire to my house). BigCorp’s CEO might go to jail in the latter scenario and not in the former, but the legal arguments I will make at court are relatively similar.
The victim of a criminal wrong may choose to either pursue a civil action, or trigger or "jump in" (constitution de partie civile) a criminal trial. While civil and criminal trials are separate in the US, the concept of a third party joining ongoing litigation is not entirely unknown ("intervention").
The procedural questions can be mostly answered by citing articles 2 to 5 of the Code de Procédure Pénale. I grouped together paragraphs of each article for ease of reading. I also used the equivalent US procedure terms for translation when I knew them.
L'action civile en réparation du dommage causé par un crime, un délit ou une contravention appartient à tous ceux qui ont personnellement souffert du dommage directement causé par l'infraction. (...)
L'action civile peut être exercée en même temps que l'action publique et devant la même juridiction. Elle sera recevable pour tous chefs de dommages, aussi bien matériels que corporels ou moraux, qui découleront des faits objets de la poursuite.
L'action civile en réparation du dommage causé par l'infraction prévue par l'article 2 peut être exercée devant une juridiction civile, séparément de l'action publique. Toutefois, il est sursis au jugement de cette action tant qu'il n'a pas été prononcé définitivement sur l'action publique lorsque celle-ci a été mise en mouvement. La mise en mouvement de l'action publique n'impose pas la suspension du jugement des autres actions exercées devant la juridiction civile, de quelque nature qu'elles soient, même si la décision à intervenir au pénal est susceptible d'exercer, directement ou indirectement, une influence sur la solution du procès civil. (...)
La partie qui a exercé son action devant la juridiction civile compétente ne peut la porter devant la juridiction répressive. Il n'en est autrement que si celle-ci a été saisie par le ministère public avant qu'un jugement sur le fond ait été rendu par la juridiction civile.
Anyone who personally suffered from the damage directly brought about by a felony, misdemeanor or petty offense, has standing for civil action. (...)
One can move for a a civil action in front of the same court and at the same time as the public action [= state prosecution]. The movant has standing for all damages (...) that are the consequence of the prosecuted facts.
Civil action as outlined in article 2 can be brought to a civil court separate from the public action. However, if public action has started, the civil action is postponed until there has been a final judgement on the public action. Public action does not mandate postponing other causes of action pending in front of the civil court, even if the criminal case decision might have a direct or indirect impact on the solution of the civil case.
Whoever brought a case in the appropriate civil court cannot bring it to the criminal court, except if the state brought a criminal action before the civil court ruled on the merits.
So, basically, a plaintiff has to choose between bringing the case to civil court or to criminal court. If the case is brought to civil court, there will be no public prosecutor. If the case is brought to criminal court, there will be one.
How to choose?
The choice of where to bring the action is mainly tactical. Some factors include:
- whether you have a choice at all. On one side, non-criminal facts cannot be pursued in criminal courts (breach of contract, divorce, etc.). On the other side, proving the "wrongful act" prong of certain cases all but requires a criminal verdict of guilty (defamation, medical malpractice, etc.)
- whether public prosecution is more likely to help or hinder. The prosecutor on an armed robbery case will generally be on the same side as the victims, but they might be neutral or outright hostile to the plaintiff in other cases. For instance, prosecutors often take the "not my problem" stance in defamation cases.
- statute of limitation: civil actions are (generally) time-barred sooner than public prosecution, and the clock on civil statutes is not reset by criminal prosecution.
Some more "cultural" notes
Each can apparently get nearly as much air time..
As hinted to in o.m.’s comment, that is much less important than it would be in the US. In the adversarial system of the US, the judge is an observer / arbiter in the trial: the parties run the show, pick "their" witnesses, ask all the questions, etc. In the inquisitorial system of France, each witness will first be questioned by the judge, not the parties, and the parties usually have very few additional questions if any (the cases that you read about in the press are high-stakes and abnormal in that regard).
...and each can ask a different verdict to the court (i.e 5 years prison, 2 years prison, and acquittal for example).
The partie civile would usually not give any opinion on the criminal punishment. Their job is to get a monetary award for their client, as fast as possible. The criminal sentencing is irrelevant to that. There are some exceptions:
- if the prosecutor requests an illegal sentence (it happens - sentencing rules are complicated), you don’t want to risk the judge granting it, because then the probability of appeal goes up drastically (which delays the whole thing)
- spending time in jail is not a lucrative activity; asking for less jail time might help with the defendant’s ability to pay compensatory damages
- comments along the line of "it would be unjust that this heinous criminal gets away with less than X years in jail" are usually unwise (why waste the judge’s time / goodwill on that? let the prosecutor do it), but may be necessary for PR or client management reasons (typically if the case is making front page news).
Note also that the judge is not held by any of the requests - they might go higher than the prosecution or lower than the defense. I know that in theory US judges might do that, but in practice it never happens. In France, it is rare but not exceptional (I would guesstimate maybe 5% of cases?).