Companions vs. trainers
It's important to distinguish Pokémon training (by those who travel from region to region) with those who simply have one or a few companion Pokémon. Almost all characters have a companion but only a proportion of each town's children seem to go off to become trainers each year.
In Ash's hometown of Pallet Town there were three other children who set off from the same class. Since his village only has a tiny population it's clear that a good chunk at least attempt to be trainers even if they don't get very far. From Ash's class, two trainers flunked out within their first year, abandoned their Pokémon and returned home and only Ash and Gary continued on. The implication is that this was a fairly normal number which is why there were (almost) sufficient Pokémon to use as starters.
The wider world
As far as how common it is in other regions, this is addressed in the prose novel Pocket Monsters: The Animation. In short, Pokémon training is a potential career and a potential source of prestige for young wannabes. Worldwide there are millions of trainers and getting into the top 10,000 is seen as a worthy feat (with the potential to make enough money to live on) and getting into the top 1000 is sufficient to make you a home-town celebrity.
Masara Town is a small town out in the boonies. Despite being legal adults, there's barely any job openings for primary school graduates there.
Due to the lack of jobs, it is a town where most children decide to become Pokémon Trainers.
In fact, most of Satoshi's seniors had left Masara Town to become Pokémon Trainers.
But unfortunately, no trainers from Masara Town ever made it into the national daily top ten charts in the newspapers or TV reports.
However, the popular children's magazine “Pokémon Comic” which boasts a readership of five million as well as the adult oriented specialist magazine “Monthly Pokémon Trainer” and its rival publication “Pokémon Pals (Commonly referred to as PokéPal)” always have a monthly Best 10,000 Trainers list.
Right near the bottom of that list, you may find some trainers from Masara Town.
When this happens, the trainers is featured on the front page in the local paper Weekly Masara Town News with a huge photo and are considered a local hero... Everyone in town throws a huge party. The only supermarket in town celebrates with a bargain sale. Outside their family home, congratulatory telegrams and bouquets are lined up, like a wreath at the grand opening of a pachinko parlour.
A man from the town named Masara Ookido ranked 931st in the national trainer rankings.
It was the first time anyone from the town had ever made it into the top 1000. Masara Ookido was more than just a local hero, he was practically worshipped as a god. In their excitement, the townspeople erected a bronze statue of him in the town square, and changed the name of the town to Masara Town.
Up until that point, it had been known as Masshiro Town.
Naturally, the new name of the town was decided by a local referendum.
On top of that, Masara Ookido was also elected as the town's mayor.