In the united-kingdom, a private prosecution is possible - although very unlikely in the case of Scotland. But extradition at the behest of a private party is not possible: only certain agents of the Crown are empowered to request it. Without the cooperation of the Crown, the most a private person could do is obtain a domestic summons or warrant. If the person named in the warrant is not in the UK then this piece of paper is not worth much. It can still be issued for a person who lives abroad, or at an unknown address, but the UK police have no authority to enforce it in another country, and it is meaningless on its own to the police of that country.
The extradition processes differ depending on the foreign country in question, but the common fact is that making an extradition request is an act of sovereign character, not something which a normal person can do on their own. Even when the UK participated in the European Arrest Warrant system, a request for a warrant could only be made by certain officials such as a Crown Prosecutor or a procurator fiscal. If they are involved in the case at all then they are likely to take it over anyway, either to proceed with it or to discontinue it.
Judicial supervision of the extradition process, or prosecution in general, also limits the extent to which a private party could use it to harrass somebody. For example, it is not a crime in the UK for someone to say they didn't like your hotel, so an aggrieved hotel owner will not get very far in their private criminal prosecution.
In a scenario where an irregular extradition had taken place - say, if the foreign police had bundled somebody onto a UK-bound plane without having gone through the proper channels - any subsequent prosecution would almost certainly be barred as an abuse of process. We know this because it has happened; see R v Horseferry Road Magistrates Court (ex parte Bennett) [1994] 1 AC 42.