How do you know they are using theor own mint ?
Discussion
You still have to fund the mint with on chain/LN and settle with LN between mints. What are you going to do only transact with yourself within your own mint?
And unless you obscure those deposited and moving funds with KYC-free/CJing, you're not anonymous or private. But at that point chaumian ecash would be redundant and you lose most/all benefits (convenience/privacy).
So to recap. If you run your own mint, fund channels with your own KYC-free btc, use your own mint with cashu, then it's private.
That's been my point all along.