Inflation stands for money becoming less valuable. When an economy faces an injection new money, the supply of money increases, resulting in inflation as the amount of goods that can be purchased with money does not change equally in the same direction.
With gold and silver mines the owner alone would not face the inflation, it would be passed on in the economy as the new money keeps changing hands.
So in the end everyone who held gold/silver before the new money entered the economy suffers from the inflation, others more and some less.
Inflation should be associated with trade, meaning those countries that actually have the silver/gold flowing through their hands get also their hands dirty. Although the economical benefits of trade and having gold mines are huge, the inflation penalty should also exist.
Lets say a Mexican mine produces some new gold/silver, some of it stays in the local trade node, but a lot moves on. Say if 10 gold is produced, and 9 is moved away by trade, only 10% of the initial increase in the money supply ends up as local inflation. In the next node, 9 gold came in, maybe 5 leaves from say the Caribbean towards Seville, 4 gold would equal 40% of the original money supply increase, resulting in countries participating and having trade power in that node suffer that amount of inflation penalty. Say those 5 that ended up in Seville than damage those who participate and have trade power in Seville, giving them also some inflation penalty. Sure, they all benefit from trade, but they also suffer from slight inflation.
Historically we had a lot of inflation also flowing from Americas to China, and from Japan to China and India and so on.
The new gold and silver coins should be a like a flow of poison running through the world economy, causing damage where they flow. Maybe Siberian and Central African states might suffer less.
It could also be worked out with events. Europeans gaining ownership of the mines either via colonial state ownership or protectorate status would result in global events that give inflation to countries that have trade power in the specific important nodes, depending on their trade power and influence.
New gold and silver entering the economy did not just result in huge Spanish inflation. In a global economy based on those two shiny metals, it slowly inflated all of the Old World.