Who should be allowed to issue a fiat currency?

On an economic basis, (that is, considering economic governance), I think it makes an interesting idea.. and local scrip has certainly appeared in both Greece and Ireland since the crisis began. It's appearance can generally be taken to be a sign of of unstable currency & prices, as well human economic ingenuity to overcome the instabilities. Of course, you can't really call it "fiat currency". When private individuals create and issue the currency... its called "scrip". In both US and latin american history, the appearance of scrip has however been a source of economic instability... because it often happened that its usefulness was limited (not everyone would accept it)...and its credibility was also limited (issuers of scrip could and did default on their obligations to honor it, causing some economic dislocation). So... overall, I think that the idea has got some potential, as long as issuing parties can be held liable to accept the scrip...and not default on their obligations on it as currency. This means that it's convertibility must be maintained. Also (and this is important)... it should not be issued on the LABOR market... because it has been used as a tool to perpetuate debt-slavery in both the US and Latin america. It should be used only on the goods markets I say. In terms of whether it might get used to hide black market transactions, I cannot speak to that.

Gold or diamonds could also be used by criminal or extremist groups. What are you going to do, ban gold and diamonds? Bitcoin could also be used by criminal or extremist groups. What are you going to do, ban encryption?

Anyone can issue any type of currency, including currency with no use value outside of being a currency, i.e, fiat. Sometimes local rules might try to prevent it, but that doesn't change the fact that it remains true that anyone can issue any currency, not just a government.   And that's the way it should be too. However, whether anyone will use a given currency is another matter entirely, and it is people's use of a currency that determines its adoption, not its issuance.   From this it can be seen that the usefulness to people of a given currency is derived from a belief that it will be needed and used by lots of other people. That explains why bulky metals have persisted for so long as means of exchange and buying-power preservation. It also explains why fiat currencies work just as well, if not better, than metallic or other physical commodity based currencies for these purposes and why government issued currencies of any kind, fiat or not, tend to become adopted over other alternatives that people might issue, even when no law or other authority tries to prevent it.   If a government requires payment of taxes in a given currency, then it necessarily creates a need for that currency, and rational users of currency for any purpose can rest assured that as long that government will collect taxes in the currency it declares acceptable, it will be used by lots of people and therefore be accepted as a medium of exchange for nearly everything else too-- it will find a very liquid market very easily. Even a very large, private corporation is unlikely to ever be able to compete with a government's currency unless the products of such a company are so universally used by a number of people comparable to those who pay taxes.  Such a condition, while theoretically possible, can occur only in two cases: 1)  If taxation -- not spending -- is a very low proportion of total exchange activity in an economy, a condition which has never existed in the capitalist era and quite possibly cannot exist as long as capitalism does.  And 2) if the power of a government to compel people to pay taxes becomes so weak that enough people doubt whether a given currency will continue to be used by others.

Looking at the existing answers, there seems to be a great deal of confusion as to what fiat money is, as well as confusion between the issue of such money and its use by people. Fiat money is simply money that is not convertible at a fixed rate into some other valuata on demand of the holder; nor is it specie. It is not necessary that there was once such a conversion that was subsequently abolished (e.g. U.S. sliver certificates). In fact, the first Chinese paper money had no backing other than the Chinese government's promise to accept it in payment of taqxes. The first Western fiat money (issued in France during the Regency of Louis XV) had the same characteristics. If people believe that a particular fiat money is not sound, they will use something else when so doing is permitted by law. A recent example was the widespread use of U.S. dollars in Russia during the 1990s. In reality, fiat money is often used because the issuing government mandates its use within its territory. Buyer and seller are generally (but not always) free to use some other means of debt settlement by mutual agreement, but if one of them insists on settling the debt in the money appropriate to the place, the other must comply. Not long ago, I was involved in an incident that illustrates this aspect of compulsion. I was purchasing an airline ticket in the Geneva (Switzerland) airport from an airline (name omitted to protect the guilty). After the ticket had been prepared, they asked for a credit card. I said that I would pay cash in Swiss Francs. They refused, saying that they only accepted cards. After several minutes of dispute, I pointed to a nearby policeman and asked if they would like to explain to him why they were refusing to accept Swiss money. The airline then declared that it would make an exception this time and accept cash. Effectively, this aspect of compulsion, as well as the aspect of acceptance for taxes, restricts the issue of fiat money to governments.

I don't know what do you mean by fiat money originated in China. So far as I know, fiat currencies do not originate by themselves. They start as commodity based currencies, then a central banks nationalise the commodity and issues currency in return. When the government runs a deficit and sees trouble brewing, it removes the backing. But by the time it is done, people start thinking of the currency as wealth and most people do not understand the importance of this backing. This started from ancient Athens and continued till the last century. Just remember whether the American public really cared when Nixon closed the gold window. Governments don't usually start printing something random currency and declare it as money. Even when the Euro started, they used a conversion rate against individual former currencies to make people accept it. In that case, people trusted the former currencies and those were perceived as backing the euro. This is what gave value to the euro.