I'll add a bit to what has been said already: you could get a voucher from ChipMixer, and exchange that with someone else (obviously someone you trust). If you do that, there's not a single entity on the planet who could know all the information linking your incoming and outgoing transactions.
So I'm quite interested in exploring this. In order to maintain your privacy from the other party, you would have to exchange vouchers rather than private keys, so I didn't know which UTXOs you ended up spending and vice versa. Given that ChipMixer now allows you to generate vouchers of custom amounts, then it is trivial to create vouchers of matching sizes.
The risk is that the other party redeems the voucher and then claims that it didn't work, or sends you a fake or already redeemed voucher. I can't see any way around this, since if you wanted to send a hidden voucher or similar to someone else then it would require the cooperation of ChipMixer, which brings us back to square one. So as you say, you could only do this with someone you trust, since I can't see a way to safely escrow or blindly send the vouchers to each other.