Who "Issues" Coins, The Mint Or The Country?

The country through their normal distribution system, usually a bank.

As a general rule, mints never "issue" coins, they merely make them when ordered to do so by the government. To qualify as a "coin", it needs government authorization to exist. That usually means it needs some kind of legislation, government regulation or decree enacted in the country of issue, either directly by the government's Treasury department (or whatever the local name for that is), or by a statutory body that the government has authorized to issue coinage on their behalf (such as a central bank).

In the case of NCLT produced by a mint in the name of other countries, it is entirely possible for a legal tender coin to be "issued" by a country, yet those "issued" coins never actually touch the soil of the country that nominally issued them. This is espeically so for countries which the mint uses as a "flag of convenience". In this case, the usual order of coinage production - government orders coins, mint produces them - is often reversed: the Mint's marketing department comes up with an idea some new NCLT coin, then shops around looking for a government prepared to legalize it. If the mint's own country refuses to do so, then they take it overseas.

Take the Australian mints, the RAM and Perth Mint. Both produce copious amounts of NCLT for the world collector market - much of which is targeted at certain thematic collectors and often has little or nothing to do with the country in question. For example, last year the Perth Mint issed a series of coins featuring the Marvel Comics characters. Marvel Comics are not Australian nor do they have much, if anything, to do with Australia, so the Australian government did not give Perth Mint authorization to make these coins. So Perth Mint went to their usual flag-of-convenience, Tuvalu, and got that government to sanction the issue of these coins. I assure you that not one single example of those coins is ever likely to be seen in Tuvalu. Yet, it is the government of Tuvalu that has, nominally, "issued" these coins.