I can answer a bit of this question with regard to U.S. law and civil (not criminal cases).
In the U.S. system, there are really two different issues, personal jurisdiction of the court and service of process on the defendant. This area is highly detailed, and whole sections of law school courses are taught on them, so my answer is necessarily general.
In the U.S., the jurisdiction of the courts (including federal courts in most matters) over a particular defendant (known as “personal jurisdiction”) is limited to residents of the state in which the court sits (though defining who is a resident can be tricky) and those out-of-staters whom under state law and U.S. Constitutional principles are considered to have a close enough connection to the state with respect to the matter at issue in the action to be fairly sued in the courts of the state. Personal jurisdiction over non-residents is known as “long-arm jurisdiction.”
Under principles of personal jurisdiction, if I’m trying to sue someone in a New York court, the analysis is the same if the person I’m trying to sue lives just across the river in New Jersey, or across the ocean in old Jersey, the Channel Island. If the Jerseyite (New or old) is being sued over a contract that called for a product to be delivered in New York or something that intentionally or foreseeably caused an injury in New York, there’s a good chance that a New York court would consider it proper to maintain a suit against the Jerseyite (either) in New York.
In many ways, personal jurisdiction is a theoretical question – based on the alleged injuries, is it fair to bring a defendant situated as described before this court. There is the much more practical question of service of process – did the actual defendant receive legally sufficient notice of this case. (Yes, I know that in many ways the issues are joined, but I’m separating them out for clarity here.)
With a natural person and in-state resident, the easiest way to serve process is to hand the person a copy of the summons and complaint (or other papers initiating the suit). Each state (and the federal court system) has complex rules about other ways a person can be served if a process server can’t simply hand the papers to him or her, and how to serve a defendant that is a corporation or other entity that may be sued. Most of the time the service can be made by any person (other than a party to the case) as a process server, but under some sets of rules or circumstances, process must be served by a marshal, sheriff, or similar official.
Under all U.S. rules of which I’m aware, to obtain service on a out-of-state person, the process must simply be served on the defendant by an authorized process server in the U.S. state where the defendant can be found.
However, when the case deals with service on a defendant that cannot be served within the U.S., it gets tricky. Most courts don’t want process servers from foreign courts running around their countries trying to serve their residents. To address these difficulties, many countries have adopted treaties regarding international service of international, such as the Convention on Service Abroad of Judicial and Extrajudicial Documents in Civil or Commercial Matters (known as the Hague Convention). Some countries have not, however, and to serve process in those countries requires a complicated diplomatic process known as Letters Rogatory. For details on serving process in foreign countries, there is a section of the U.S. State Department website that explains some details.
Once you have personal jurisdiction over and have properly served a defendant, the action against the defendant can proceed. If you win the action, the next challenge becomes enforcing the judgment. Courts only have jurisdiction over assets in their geographic area, so to satisfy a judgment, you must get the judgment entered in a locality where the defendant has assets.
Under the Full Faith and Credit clause of the U.S. Constitutions, all U.S. states are required to enforce judgments of sister states, so enforcing a judgment against property in another U.S. state involves the relatively simple (but still awkward and procedurally challenging) effort of registering your judgment in the state where the assets are located, and then enforcing that judgment under the rules of that state. (The enforcing state, however, can sometimes re-examine whether the original court had personal jurisdiction and proper service over the defendant if the original court did not do so).
When you are trying to enforce a foreign judgment, though, it can get really tricky. The foreign court will often look at all procedural aspects of the case and the circumstances under which the judgment was obtained and decide whether to enter a similar judgment under the judicial principle known as comity. Although I would generally say that the courts of most countries under an Anglo-American judicial system (U.S., U.K., and most former British colonies) will respect the judgments of similar courts in non-controversial cases, beyond that just about all bets are off.
If you couldn’t tell, I just finished writing a brief on this very subject. Obviously, this is a very complex area of the law, and you should consult a lawyer in your jurisdiction (and perhaps the jurisdiction of the person you wish to sue) for any specific questions.