Azkaban is currently the only wizarding prison in Britain
Once the International Statute of Secrecy had been imposed, the Ministry of Magic felt that the small wizarding prisons that existed up and down the country in various towns and villages posed a security risk, because attempts by incarcerated witches and wizards to break out often led to undesirable bangs, smells and light shows. A purpose-built prison, located on some remote Hebridean island, was preferred, and plans had been drawn up when Damocles Rowle became Minister for Magic.
Rowle was an authoritarian who had risen to power on an anti-Muggle agenda, capitalising on the anger felt by much of the wizarding community at being forced to go underground. Sadistic by nature, Rowle scrapped the plans for the new prison at once and insisted on using Azkaban. He claimed that the Dementors living there were an advantage: they could be harnessed as guards, saving the Ministry time, trouble and expense.
Pottermore - Azkaban
The implication given is that other countries would have their own wizarding prisons, but due to the books focus on England, we never learn hear about them.
The only other prison that we do hear about is Nurmengard, but it appears to be no longer used a regular prison.
They say “For the Greater Good” was even carved over the entrance to Nurmengard.’
‘The prison Grindelwald had built to hold his opponents. He ended up in there himself, once Dumbledore had caught him.
Harry Potter and the Deathly Hallows - Chapter 18
So to answer your questions:
There are other prisons in the magical world, just not in Britain.
People from Britain are sent to Azkaban. People from other countries are not.
"Minimum security" setup prisons for lesser crimes existed prior to the adoption of the Statute of Secrecy.