The trial has commenced within the Vatican of, former Archbishop
Vienna Convention on Diplomatic Relations of 1961 which has been ratified by virtually every country in the world. Under the Convention Diplomatic Immunity is unambiguously protected
Article 29: The person of a diplomatic agent shall be inviolable. He shall not be liable to any form of arrest or detention.
Article 31: A diplomatic agent shall enjoy immunity from the criminal jurisdiction of the receiving State.
Therefore as a diplomat the normal rule is that Weslowski could neither be arrested, questioned nor prosecuted in the Dominican Republic regarding his alleged crimes. However as a diplomat of the Holy See he is subject to the criminal law of the Vatican State and that criminal law extends to prosecuting him for offences committed abroad whilst acting as Nuncio. Diplomatic Immunity protects a diplomat from the criminal law of the country he is sent to but it does not protect him from the criminal law of his own country which is why Weslowski is able to be prosecuted in the Vatican.
Had Weslowski not been a Papal Nuncio but instead been an ordinary Bishop in the Dominican Republic then the question of Diplomatic Immunity would not have arisen and he could have been tried there, "Ordinary" Bishops, Cardinals etc who are not Papal Nuncios are not protected by Diplomatic Immunity. On the same basis since they are not citizens of the Vatican State they are not be subject to the laws of the Vatican State except when they are physically there.
The Investigation of the allegations against Weolowski will have been carried out by the Vatican Gendarmerie who are all former officers in one of the Italian Police Forces and who will follow standard Italian Police procedures. The trial itself will similarly be conducted according to the rules of Italian criminal procedure which have been adopted by the Vatican since the Lateran Treaties of 1929. The Judges in the case will be lay Judges trained in Italian civil law. If convicted Wesolowki can be sentenced to imprisonment and under exisiting agreements between Italy and the Vatican he could serve any prison sentence in an Italian prison.
The trial Wesolowski has already faced, was a separate trial under Catholic Canon Law which is a code that applies to Catholic Priests and Ecclesiastics throughout the World and which relates to whether they have broken the rules which apply to them as Priests. An offence under Canon Law may well not be an offence under the Civil Law of the country in which it took place and the penalities under Canon Law are purely religious penalties. In the case of Wesolowski the penalty imposed by the Canon Law trial was that he was stripped of his priestly status and laicised, in England often referred to as "being de-frocked".
As already mentioned if Wesolowski had not been a Nuncio and was not covered by Diplomatic Immunity he would have been dealt with by the criminal courts of the Dominican Republic and not the courts of the Vatican; however he would still have had to be dealt with separately under Canon Law to decide if he should be laicised
There are therefore 2 separate trials because there are 2 separate legal systems involved. There is the Vatican State legal system, which is specific to Wesolowski and the small number of priests and Ecclesistics who are Vatican State citizens, and there is the Canon law system which applies to all Catholic Priests everywhere