Since I don't have any lawyers on hand, I'm not qualified to design a private system so I don't feel like making one up[1]. But since I remember how bank transfers work right now (hint: they go through the government), I can say people claiming they have some right now are wrong.
[1] for instance, I don't remember the legal status of mailing cash through USPS, but pretty sure it's different privacy-wise even though it's "the government"
[1] for instance, I don't remember the legal status of mailing cash through USPS, but pretty sure it's different privacy-wise even though it's "the government"