"Agents of Interpol" have been a media theme for many years, but they're a media invention. There are no Interpol agents.
Interpol is an international association of governments with the mission of assisting and coordinating law enforcement efforts among its members. It has no police powers itself.
Each member country designates a point of contact, a liaison to the other members. When some investigation or prosecution requires the assistance of another member country, the law enforcement agency needing the assistance contacts the liaison for its government. After reviewing the request for propriety and proclivity for giving rise to international incident, the liaison contacts his opposite number in the other country. That liaison, on getting approval, forwards the request to the appropriate law enforcement agency in his country.
Roman Polanski was the focus of an Interpol operation when he went to Switzerland to accept an award. Polanski has been living in France since the late 1970s to avoid a prison sentence for a crime in California. France has refused to extradite him. Switzerland was more inclined to entertain extradition.
I imagine the process went something like this: on hearing that Polanski was going to Switzerland, someone in the California attorney general's office (or the Los Angeles district attorney, or maybe LAPD—this is all conjecture) sent a message over the National Law Enforcement Telecommunications System (NLETS) to the Interpol liaison at the FBI, asking for help from the Swiss cops. The FBI sent the message to the Interpol guy in Switzerland, who got in touch with the appropriate Swiss agencies. At some point, the two agency reps in each country were talking to each other directly, with the Interpol reps staying in the loop.
The Swiss cops arrested Polanski, but after their courts reviewed the case, they elected not to allow Polanski to be extradited. Polanski flew back to France, where, so far as I know, he remains.
Few countries will permit anyone but their own officers to exercise any police powers in their country. Foreigners can ask another country's police to arrest someone or interview a witness, but the host country can tell the requestor to pound sand, too. Interpol helps grease the skids, ensuring that the request won't ruffle any feathers and that proper protocol is followed. It's an association with no officers or powers of its own. There are no international cops. When a cop leaves his home country, he becomes a private citizen.
More questions on law enforcement and the police: