If the "functioning government" in question is a democracy, Pokémon battles are such an integral part of culture in the Pokémon world that there's no way they would shut them down. (Except for Fiore, but the wild Pokémon there still battle. Otherwise the Rangers wouldn't be able to call on any useful Pokémon.) If it is a kleptocracy, there's no way the number of industries involved in Pokémon battles are going to let the government shut down. If it is a monarchy or aristocracy, they still want to preserve popular opinion so that the commoners don't rise up and overthrow them. Even if it's a self-destructive Nazi-style dictatorship that will probably be gone by the end of the decade, Pokémon battles do provide effective bread and circuses for the masses, and why outlaw them when training Pokémon provides effective weapons of war?

tl;dr your opinion is wrong

But I'm pretty sure the gist of it it 'if we catch you abusing pokemon we shoot you on the spot and also we've got a network of assassins to kill anyone who does crime'.

Ah, here it is.

Does anyone know what the PAL region is?

PAL stand for Pokémon Awakens Love. It's a region that supports all six regions known to the Pokémon World. Some people like to go to PAL because according to the media, it was very beautiful. It was very sunny during the spring and summer, many colourful flowers and lush green trees and grasses in the many natural habitats PAL has and all their water bodies are crystal blue, sparkling and clean. The bushes were also bursting with Berries for Pokémon to eat. The general environment a paradise for Pokémon; hence the name. If a trainer is caught abusing Pokémon within this region, it's an immediate death sentence, as PAL takes the welfare of Pokémon, whether wild or trainer, very seriously.

With this in mind, pollution is also considered a crime here. If you are caught tainting the environment in anyway, you'll be sent to jail. And let's just say landing yourself in PAL's Jail is never a pleasant thing…..it's much worse than the jails in other regions.

Trainers are expected to do everything they can to protect their own Pokémon, which is why all the trainers living in PAL have to learn how to fight in one way or another without their Pokémon. PAL contains all the Pokémon in the six regions, and also, sometimes, even Legandaries flock here to relax in this paradise.

Let's not forget those that protect and run PAL. Officer Jennies around the region are nothing compared to them. They are called "Hunters", people that have proven themselves through rigorous examination to be an elite member of humanity. If you can pass the Hunter Exam, or "Exam from Hell" as PAL people may call it (since out of 1,200 applicants, only about 10 pass and get their license, and some actually died in the Exam), you have many benefits, such as 95% of all public facilities can be used at no cost. Furthermore, all such services are provided at first class, unrivalled freedom to travel around the regions, almost no legal responsibility for murder, and so much more.

Most people from the G-Men members are actually Hunters provided by the PAL's Prince and Princess, which is why they are considered formidable. PAL is a principality, which means they were ruled by a Prince or Princess.