I haven't actually checked network requests on all or even most of them, if this is what you're asking, but given how (positively) anal the Linux community is a

I haven't actually checked network requests on all or even most of them, if this is what you're asking, but given how (positively) anal the Linux community is about tracking and how I've not actually heard about anything other than the technological feasibility of Canonical spying on users, I'm rather confident to say that none of the popular ones do. So, to throw about a few names: Debian, openSUSE, Trisquel, Ubuntu, Fedora, Slackware etc. Or to put it differently, the distros of which I'm aware that they track users: Android and Red Star OS.