The privacy properties for P2P transfers between consumers are identical to those with merchants: the exchange can require the receiver to provide information about themselves, but the payer remains anonymous just like for purchases at a merchant. And yes, it involves spending (burning, if you prefer) the old tokens (we prefer digital coins) and issuing (blindly signing) fresh coins. Afterwards, the fresh digital coins are indistinguishable from those issued during a regular withdrawal from a bank account.
Would it be technically possible to transfer tokens to another user directly, without an exchange acting as an intermediate?
My assumption is that this is possible but not supported by default, since there’s no way to guarantee (without using an exchange) that the giver has really deleted the tokens they sent to the receiver, and will not spend them before the receiver can (which would render the receiver’s copy of the tokens unusable).
But assuming someone implements this in software, would it be possible to make direct transfers like this while relying entirely on trust? Or are the tokens somehow tied to the wallet that has first received them?
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Right, because you can issue coins as a "purchase" of the coins being transferred. That makes sense, but still doesn't address the concerns around the system being designed to support negative interest rates and withdrawal rate limits.