That means you register your inputs and blind outputs with one Tor identity, but your unblinded outputs with another Tor identity.
Indeed, the coordinator knows nothing about who owns what. Although, it isn't so simple to comprehend at first sight, the two parties can hide their outputs (hence why "blinded outputs") using blind signature. And according to a BlindCoin's paper, older bitcoin mixers such as Bitcoin Fog, Bit Laundry and Blockchain.info hadn't used it.It's also explained in the wiki:
Quote from: https://en.bitcoin.it/wiki/CoinJoin
Don't the users learn which inputs match up to which outputs?
In the simplest possible implementation where users meet up on IRC over tor or the like, yes they do. The next simplest implementation is where the users send their input and output information to some meeting point server, and the server creates the transaction and asks people to sign it. The server learns the mapping, but no one else does, and the server still can't steal the coins.
More complicated implementations are possible where even the server doesn't learn the mapping.
E.g. Using chaum blind signatures: The users connect and provide inputs (and change addresses) and a cryptographically-blinded version of the address they want their private coins to go to; the server signs the tokens and returns them. The users anonymously reconnect, unblind their output addresses, and return them to the server. The server can see that all the outputs were signed by it and so all the outputs had to come from valid participants. Later people reconnect and sign.
Similar things can be accomplished with various zero-knowledge proof systems.
In the simplest possible implementation where users meet up on IRC over tor or the like, yes they do. The next simplest implementation is where the users send their input and output information to some meeting point server, and the server creates the transaction and asks people to sign it. The server learns the mapping, but no one else does, and the server still can't steal the coins.
More complicated implementations are possible where even the server doesn't learn the mapping.
E.g. Using chaum blind signatures: The users connect and provide inputs (and change addresses) and a cryptographically-blinded version of the address they want their private coins to go to; the server signs the tokens and returns them. The users anonymously reconnect, unblind their output addresses, and return them to the server. The server can see that all the outputs were signed by it and so all the outputs had to come from valid participants. Later people reconnect and sign.
Similar things can be accomplished with various zero-knowledge proof systems.
It's nice to read all these proposals and papers which were written overtime, trying to make it better. Bitcoin has, undeniably, come a long way in such short period.
However, I think this requires a small edit.
Quote from: https://en.bitcoin.it/wiki/CoinJoin
Wasabi works thanks to a CoinJoin coordinator (run by zkSNACKs Ltd., the company that is sponsoring the development of Wasabi) who cannot steal from, nor breach the privacy of the participants. The company makes its income by taking a fee (0.003% * anonymity set) from each CoinJoin transaction.