Run that against a normal flow and it falls apart. The exit point on chain A is a transaction with a sender, a receiver, a contract, an amount, and a fee, all public. The entry point on chain B is another public transaction, and the two are linked by a server-side pairing that anyone with the deposit and the recipient address can correlate. Privacy, in the sense a routine user means, comes from breaking those links, not from the bridge being non-custodial.
So the honest version is: a bridge gives you custody-free movement and nothing more. If the chains on either side are transparent, and the operator can see both sides, the linkability is the same as a CEX withdrawal-to-deposit pattern. Three addresses that share a common input, a round-number amount, or a 10-minute window cluster trivially. From there, any KYC'd off-ramp downstream finishes the job.
What changes the picture: relayer sets that batch across many users, destination addresses generated per-deposit with no reuse, and a fee path that adds noise. Used together they raise the cost of the obvious heuristic. None of them make the on-chain trail disappear, and a chain analyst with both endpoints still has the receipts.
For a routine user the practical test is simple. Send a small amount across a [crypto bridge](), look up the destination address on the receiving chain's explorer before funding it, and see whether the same address, the same timing, and the same amount are visible from the other side.
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