I'm asking again about an other hypothetical scenario. Supposing that there are two countries having extradiction deals and treaties with one another (let's suppose for simplicity sake, two EU countries) A, and B. Let's also suppose that a person, living in country A commits defamation against another person, living in B, by committing, for example, slander (at the moment of the defamation the victim is always in B). If the person whose reputation has been ruined and spoiled decides to take the issue via legal means and thereby to sue the perpetrator, will the case be pursued in the country in which the act has been committed or in the country in which the victim has endured the slander (the thing might occur on the internet for example, and the two places do not coincide)? Note that EU countries have similar laws for this thing, but I ask also for other couples of countries. P.S I don't know whether extradition has something to do with the issue I'm asking about
So in the EU (or in an Intra-state case in the United States) the trial will typically be tried in the state were the damage was done. A recent example of this is the ongoing "Covington Catholic High School" defimation case in the United States, where the incident that lead to the alleged defimation took place in Washington D.C. (D.C. would be the "state" for puropses of jurisidiction), the parties to the defense exist in different parts of the nation but typically news media heavy states of New York, Califorina, and D.C. Never the less, the plaintiff(s) live in state of Kentucky. Because the alleged damages to the plaintiffs will most affect them in the communities where they live, it was ruled that Kentucky will hear the case's arguments rather than the NY, CA, or DC and that Kentucky law will be the laws considered broken.
I believe the EU will do the same, but my one caveat is that the U.S. and most of the EU do not use similar legal frame works and part of the jurisdiction decision will change the jury pool with respect to the opinoin of the locality. In this case, it was vital that the victim's community judge the merits of his case and what is damaging as Kentucky citizens does not have the same common ground of "this is morally wrong" as NYC, CA, and DC and are much better suited to understand the victim's hardships. With exception to Great Britain and Ireland, most EU nations do not use Juries to decide legal cases.
That said, the court in which the case will be filed is that of the victim, B, and provided the EU has standard extradition treaties with all members, the defendant will be required to argue the case in nation B.
With respect to Extradition between two nation-states that are not joined in some confederation (EU) or Federation (U.S. States), extradition between two nation typically requires the requesting (making the charge) and responding (home of the defendant) nation to both have an Extradition Treaty and for the Responding Nation to have similar crimes and punishments to those charged and faced by the Requesting Nation and that the Requesting Nation show sufficent evidence in the Responding nation's court to find probable cause for charging the defendant as if the crime was comitted in the Responding Nation's jurisdiction (for purposes of this hearing, the location of the crime remains, but is treated as if the responding court had jurisdiction to begin with). Finally, once granted and in the custody of the requesting nation, the requesting nation may not file any new charges against the defendant without a separate extradition request.
In the case of defamation, generally the United States is generally more liberal on what is a defimation case than the EU and many people seeking defimation against a U.S. citizen would file suits in British Courts to get around the U.S. courts. To stop this, the U.S. made a law that basically refuses to honor any damages for defamation to other nation's courts unless the case could also win in the United States. This also goes against them, as the Canada and many other countries will not extradite anyone who is being sought for a crime that could result in capital punishment (though the United States has found that a mandatory death penalty sentance is unconstitutional, so as long as the prosecution promises not to seek the death penalty in the case, the extradition can happen (US trials do not come with an automatic sentencing upon finding of guilt. The sentencing phase is typically separate part of the trial and is done usually about a month or so after the conclusion of the Jury Phase, where evidence to aggravating and mitigating factors are considered. Guilty on a case which is eligible for the death penalty is not automatically a sentence to death, so it works in this case.).
Additionally, if no extradition treaty exists, then the responding nation may still decide to extradite the defendant, but this is considered a diplomatic gift if done in this manner. They are not obliged to extradite someone no matter how nice the requesting nation asks or threatens (the U.S.-Afgan war (2000-present) started because the U.S. followed up on their threats for a non-treaty extradition of Osma bin Laden). If the Responding nation does choose to extradite, the requesting nation is not obligated to honor any quid pro quo, but because the lack of a treaty means that the two nations are typically not on friendly terms, the quid pro quo is usually helpful in securing the prisoner. While not an extradition, spy swaps commonly depict a similar shady situation where two rival nations give a spy they captured back in exchange for the return of their own captured spy. The classic foggy bridge is based on a real bridge in Berlin where the U.S. and Soviet Union would frequently make the exchange during the Cold War as the bridge spanned a river dividing line between east and west Berlin and had enough open space on both ends that both nations could make ensure that the other nation was up to shenanigans. The film "Bridge of Spies" shows a real life example of this exchange, where Tom Hank's character was able to Bargin for two U.S. Prisoners behind the Iron Curtain in exchange for convicted Soviet Spy Alger Hiss, who was kept alive specifically for this purpose (the scene is not on the real bridge as it was impractical to shoot for the film, mostly due to the period piece nature of the film.).
A real life case of this was Julian Assange where the nation hosting him in their Embassy in Britain finally got sick of their guest and allowed the British Police to come in and arrest him on charges of skipping bail (related to extradition hearings from the U.S. and other countries related to various criminal matters).
The question has some terminological errors which make it difficult to answer. Defamation is not necessarily a crime, for example in the US it is a civil matter where A may sue B because of the damage arising from the statement. It may be a crime in some jurisdictions (Turkey, Indonesia) – this article surveys criminal defamation in a number of countries. The intermediate category "misdemeanor" in English and American law doesn't trivially translate to other systems, for example
Crimes in French law are the most serious offenses, punishable by death or prolonged imprisonment. A délit is any offense punishable by a short prison sentence, usually from one to five years, or a fine. Contraventions are minor offenses.
Since the question seems to be the most about (slander) defamation, then the question is whether the person A is being prosecuted by the state for violation of that state's laws against defamation or insult directed at B, and possibly imprisoned? For that to happen, A must be subject to the laws of that country, for instance you have to be in Turkey to violate Turkish law. There is a complication of extraterritorial jurisdiction, where a country may claim jurisdiction over people outside of that country (typically, citizens of that country, who may be prosecuted for offenses committed while abroad). Either way, to be tried under Turkish law, you have to be in Turkey, which implies that if you are in Germany, you have to be extradited to Turkey. A complication of that is trial in absentia.
I think it is not practical to look for a general global answer or even a European answer, because you have to know whether the countries in question have extradition treaties and they may have laws prohibiting extradition for certain offenses. The status of the defamed person matters substantially in terms of criminality (insulting a public figure can add years to the sentence, but is likely to make the crime a political and thus non-extraditable offense).