That was my assumption.

Your comment took me a little by surprise actually because there's no actual need for the internet in SSH and in fact I probably do 90% of my SSHing on local networks. But I guess your usage differs significantly?

For desktops, yes, but my default assumption for a phone is that it's on a mobile network

I don't think that is a fair assumption because it's very common for people to set up WiFi on their phone. In fact you'll often find some platforms will not download OS updates or warn you against downloading larger "apps" when not connected via WiFi.

A mobile network is still mostly a fallback "if there's nothing better available", and IIRC all current smartphones prefer a WiFi link over a mobile one (if one is up). Perhaps with 5G, we might get to the point where WiFi is too much hassle to set up...

That's the usual arrangement, yes: phone - WiFi AP - ethernet switch - desktop; no internet needed. SSH is not some only-works-in-the-cloud magic, just a TCP/IP service; as long as you can access the server, it is cloud-agnostic.

I do use this arrangement with split DNS: inside the local LAN, the desktop's DNS resolves to the desktop directly; outside, it resolves to the gateway's external port, whence it's NATed to the desktop. SSH from anywhere :)