They know an IP address. But, without a ridiculous amount of work, not my IP address.
But anyway, the thing is that I do know that I'm under surveillance at least some of the time. So I might as well assume that it's all of the time. Because it's hard to know.
And sadly enough, I won't even speak freely with my wife, unless she's turned off her cellphone, or put it in the utility closet or bathroom, with the water running.
For example, two decades ago I was ~comfortable driving a couple hundred km to a mall, and buying a gift card for cash. Or mailing cash in the mail. I drove back roads, avoided toll bridges, and didn't carry a cellphone.
The moral is that being ~anonymous in meatspace has become very difficult.
So yes, I'm no longer comfortable assuming that such steps would guarantee any privacy. And yes, I'd also worry that at least parts of the process would be recorded. And the harder I tried to obfuscate, the more likely it'd raise suspicions.
I do worry about using a VPN. But that's not too uncommon where I live, so hopefully it's not a huge flag.
I neglected to say that I was doing that with the expectation of ~anonymity. Using those gift cards to lease VPN services and VPS. Mailing cash to people selling Liberty Reserve and Pecunix. And later, Bitcoin.