The comments on SingerOfTheFall's answer is on the right track but not quite got it yet.
Yes, when new coins of lower purity are introduced to circulation, all coins of that type will lose value. It's due to the people using the coins having less trust in said coins.
While that is the first step to understanding the scheme, it goes much deeper.
When Milone Trading company starts collecting the Trenni Silver Coins, they are not doing so because they want them to be recycled in exchange for exclusive deals. First, you have to think about why that particular country would be lowering the purity. It's because the country of Trenni doesn't have the silver to continue making coins of such high purity.
Yes, in the short view Trenni can make more coins, but this also means that outside of their own borders, the coin will be worthless. In addition, within their borders, the coin is not valued on how much silver is in it so much as it is by how much the government says it is worth.
For example, let's say you have one pure 1 gram gold coin. Let's also say these pure gold coins are the common currency in your country. That means even though it is just 1 gram of gold, the government can say that the 1 gram is worth 2 grams. This is something called seigniorage. Now, this only works to a certain degree, which depends on how much faith the people have in the coins purity and the government manufacturing the coins. So, it is in the country's best interest to keep purity high. However, when they decrease the purity, while it will make a lot of money really fast and put more in circulation, it does decrease the faith that the people have in the coin. Meaning that even with a small decrease in purity, you may see a drastic increase in how many coins any given item is worth due to people not having faith in the purity of the coins. Therein lies why it isn't a good idea to reduce the purity so drastically.
If the Government is reducing the purity, you can assume this means their funds are getting low. Meaning that they would be more than eager to trade exclusive deals for coins they can recycle. In that, the commenters were correct. But, think. If Trenni were to suddenly come into a large amount of silver they wouldn't have to recycle all these coins into lower value ones. They could slowly reintroduce them into circulation and therefore solve the problem of having such low funds. That is why this deal was so valuable and the land of Trenni would be willing to sacrifice quite a bit for the older coins.
Also, Milone's goal was not to trade for deals. No, instead they traded all the older higher purity coins for land, large area mining rights, tariff control, and other privileges generally held solely by the government. Meaning they could move their goods without paying tolls, resale the land at a higher price, and sell individual mining rights for a small area to smaller companies for a higher price per acre. For that matter, they could even further use the land and mining rights for future income by leasing the land and mining rights for certain areas in segments. Though it would be a little longer before you made your money back, you would in the long term have a potential to earn much more than you would just by selling it.
Now considering they were able to pay Lawrence back the next morning, I would assume they resold all or at least some of the privileges they acquired, which is the least risky way to play it since if they were to lease the land, there is no guarantee that people will be willing to lease it. That leads to the risk of actually losing money. It's my guess they sold the risky investments and kept the privileges that would lead to the saving money in the future as their portion of the profit.
Or there is a second option in which they could have kept all the assets and privileges they gained and gave Lawrence Trenni coins out of the Companies pocket based on the estimated value of the assets they'd acquired. That way, Milone Trading still has the potential gain from the mining rights and such.
And then there's yet another option where they sold just enough of the assets to make sure they hadn't suffered a loss in addition to the amount needed to pay Lawrence. This, in my opinion, would be the smartest play considering it would mean there is no risk of loss and their debts are settled. In addition, they would have the leftover assets to do what they wanted with them. In theory, this option is almost no risk with high potential gain.
Now to tackle why Medio would be spreading words of raising purities, and truthfully, compared to the rest of all this, it's quite simple. If the community believes the purity is going to raise, then they will begin getting rid of older coins and trying to acquire these newer "thought to be more valued" ones. This gives Medio to act as if they are a third uniformed party and give the people newer coins in exchange for older coins. Likely they had the same goal as Milone, this was just their way of collecting the coins. Unfortunately, the direction they chose to collect the coins also requires them scamming all the people that they tell into having the newer less pure ones. That part of it really is quite simple.