Yes when you have monopoly you can't get paid in other currencies, that is correct. However, it would not end up owning most of the economy, since the point of the plan is to give away the money for charitable purposes (which might not only include money to the poor of course but also other things the effective altruist movement find important).
I don't think that you'd have to be the government for this to work. You'd have to strike a deal with the government, that is true, but you would not have to be the government. You'd be very powerful, that is true, but there have been other organizations which have been very powerful without actually being the government - e.g. labour unions in some countries. Of course, they have had lot of contact with the government - sometimes friendly, like under social democratic governments, sometimes hostile, like under Thatcher.
The power of such "states in the state" is of course upheld by the legal system. Social democratic governments tend to write laws that support the union power, whereas conservative governments write laws that undermine them. See Gunnar's post on how this system would be upheld by the legal system too.
So in order for this system to be accepted, you need political support, for which you in turn, in a democracy, need electoral support. This does not, however, mean that you should form a political party for this cause. Instead I think that a better idea would be to work it out on your own, and let the parties come to you, as it were. That way the distinction between the NGO that takes care of this system and the government would be upheld.
In the last few years we have seen two interesting revolutionary ideas on how to change the monetary system. The first is Bitcoin: the most well-known peer-to-peer currency. It has been wildly debated recently and I won't go into the detail of allegations of use in criminal activities etc (for one thing, I don't know much about it). My interest is rather in the money creation part. The people who run the Bitcoin software are rewarded for their work with new Bitcoins - a process called mining. Now the pace at which new Bitcoins are mined is limited, which means that Bitcoin creation is a zero-sum game: the more one miner contributes to the Bitcoin software, the less Bitcoins other miners get. Unsurprisingly, this has led to an arms race: miners spend nearly as much on running the software as they get back in form of new Bitcoins.
The second idea is the Chicago Plan, which was debated already in the 30's, after the great crash of 1929, but which recently was resurrected by Michael Kumhof (senior economist at IMF, of all places). The central idea of the Chicago Plan is to abolish fractional reserve banking - the system by which private banks in effect create money out of thin air. Instead of lending out most of the depositors' money, banks would effectively have to let them stay in the bank.
Instead money would be created by the central bank/government, a process that would generate a massive seignorage for the government. According to Kumhof, it would also have other beneficial effects, such as killing off the "boom-and-bust"-cycles which he thinks fractional reserve banking are mostly responsible for, and diminishing the wasteful parts of the financial sector.
Kumhof ideas' have not been well received. Overall, it is remarkable how little reform there has been of the financial and monetary system given that the world had a major financial meltdown 2008 (and was close to an even greater one, in my understanding). Governments won't challenge the financial system radically in the near future, that's for sure.
Instead radical reforms can only come from private hands. Let us now compare the two ideas. In the Bitcoin system money is created by private hands, but in wasteful ways, which effectively means that there is very little seignorage. Under the Chicago plan, money is created by the government in much more efficient ways, which leads to a large seignorage. Now my idea is to take the best part of both of these ideas: let a private player - more exactly, an altruistic organization such as CEA - produce the money centrally, Chicago plan-like, and let the seignorage be used for altruistic purposes. (Of course, there would be some costs of running the system, but if the system was sufficiently large, these would be negligible in relation to the seignorage.)
If the altruistic organization that did this had a sufficiently good reputation, chances are greater that people would trust the system. Of course, it would try to stop the currency from being used to launder money, drug trade etc.
Generally, people would be suspicious of private currencies where the central authority collected a seignorage, but if this seignorage was used for charitable and other altruistic purposes (and people really trusted that that would be the case), this would, I hope, be less of a problem.
What do you think? I'd be happy to get comments from people who know more about the Bitcoin system, since I don't really know it (though I find it interesting). Perhaps there is some info concerning Bitcoins that tells against this proposal; if so, I'd be interested in that.