Ask HN: Does Claude AI run locally?

I have a variety of devices of different ages, and I’ve noticed that Claude specifically appears to run locally. No I’m not referring to Claude Code, the CLI tool. I’m talking about the website in any browser.

I have an old Toshiba Satellite for example, when using Claude my fans ramp up and the laptop heats up like it’s a bitcoin miner. No other AI chat client website causes this. This is not to say it doesn’t ramp up the CPU of my other devices which it certainly does.

Has anyone else noticed this? I try searching to no avail. Everything I see says it does not run locally but my user experience, and resource monitors, says otherwise.

Edit: I ask because why the fuck am I paying $20 when I’m doing the computing? Why are any of us paying?