There are only two methods of exchanging goods:
2) An externally managed and regulated medium of exchange.
There are some historical examples of "bottom-up" unregulated currencies. Apparently there is, or was, one in Somalia for a time, with old banknotes that had been issued by the now-defunct government redenominated and used as tokens in some of the urban markets. They had exchange value but only by consensus - there was no external authority guaranteeing them. Another example are some of the exchange mediums used by small non-industrialized societies, typically shells or stones or the like.
But those fail at larger scales because they depend on social pressures (reputation, social mores and sanctioning) to enforce value. Among strangers, which brings us back to your two options. Even distributed cryptocurrencies are regulated by the systems that implement verification and settlement.