Well, most or all industrialized countries - including the United States - allow “competing currencies”, in the sense that you are perfectly free to enter into a contract with somebody to buy or sell or lend or - well, do anything really - in any currency which you and the contract counterparty may please.
And people do this quite a lot on the financial markets. Just not at the level of consumer transactions, because it mostly isn’t very convenient at that level.
The point about legal tender laws is not that everybody has to use the legal tender currency, but that every creditor must accept the legal tender currency - if offered - in settlement of his debt. Or, to put in another way, debtors have the right to pay their debts in legal tender currency.
Designating a legal tender currency doesn’t prevent people from entering into transactions denominated in other currencies. So, for example, if you and I are both resident in the United States, there is nothing to stop us entering into a contract whereby you will sell me a cup of coffee for (say) one euro. And you can, if you wish, refuse to sell me the coffee for the US dollar equivalent of one euro. Because a contract to sell does not create a debt, it is unaffected by legal tender laws.
Similarly, we can enter into a contract under which you lend me (say) 1,000 euro and I promise to pay it back, with 5% interest, in one year’s time. If, when the year is up, I turn up with 1,050 euros and offer it to you, we’re good. If I offer you US dollars, you’ll have to accept the dollars, but I’ll have to offer enough dollars to allow you to buy 1,050 euros, which is what I owe you.
By exension, there is nothing to stop us entering into contracts denominated in a currency of our own devising, or devised by some (private) third party, or denominated in acorns or pebbles. We just have no reason to do so. If the US stopped issuing its own currency and created a free-for-all, it’s very unlikely that we would do this; we’d more likely use euros, or Canadian dollars, or Chinese renminbi.
Wikipedia suggests that some countries outlaw the use of foreign currency, or require a licence to transact in foreign currency, but I’m pretty sure that the US is not one of these, and in fact I don’t know of any major country which does this.