No need to use VPN/TOR when mixing coins.
Not at any stage in the process? I am surprised.
It is not necessary, because the problem is taht your coins now are linked to your ID somewhere, right? An exchange probably, which you did KYC. As exchanges addresses are known, those coins are linked to an exchange (in the blockchain), and that exchange has your ID. so they are linked to you.
Once you get new coins from the mixer, those coins are not going to be connect to you in any way in the blockchain. Blockchain does not store any IP related data.
There might be some exceptions if you are ultraparanoid (and don't know any now), but it is not necessary for average users who just want to get new clean coins.
It is also important to remember that you shouldn't mix ALL of your coins, because in the future we don't know if those coins that came from mixers will be "tainted". Unless you really need to do so.
This seems somewhat ironic, as the tumbling process seems to be promoted as some form of sterilization.
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Well, not that ironic.
You will have "sterilized" coins. Everyone will know those coins came from a mixing service.
Why did you sterilized it? Did your money came from ilegal activities? you cannot prove it, no one can, because they are sterilized. Authorities may aask: What are you hiding?
Some exchanges do not accepts coisn that came from mixers or casinos. However, there are ways to bypass that check.
But we don't know how will authorities deal with mixing services in the future, neither how will each country deal with that.
You can watch this video from antonopoulous about it:
Bitcoin Q&A: Blacklists, taint, and wallet fingerprinting