None of the 5 places I have worked is this possible, but they are also all highly regulated industries. Firewalls block virtually everything by default.

Fair, but I assume everything on my work laptop is key logged. Surely they would notice Claude phoning home from my company laptop? I suspect a network rule to look for that traffic is trivial?

My employer doesn't specifically block this stuff, but does put up a warning when you visit it to review our AI usage policy. There isn't detection for using things in ways we shouldn't, but they have an audit trail and can review it if there is suspicion.