Thank you for your inputs folks, and for the technical details some of you have forwarded.
Here is an example of the internal request made by a browser to include an image. This example is from a link to the image at
http://www.wellho.net/pix/ranpic.jpgRequest to the server once connected:
GET /pix/ranpic.jpg HTTP/1.1
Referer information included with the request:
http://www.firstgreatwestern.info/Browser description string included with the request:
Mozilla/5.0 (Macintosh; Intel Mac
OS▸ X 10_15_7) AppleWebKit/605.1.15 (KHTML, like Gecko) Version/15.6.1 Safari/605.1.15
Servers which allow hot linking have - as described earlier - software checks on them to see if they consider the requests genuine ones rather than churning out loads of responses to automata, and my conclusion is that some combinations of the referer and the browser (and perhaps the IP you're browsing from) are rejected.
I'm pretty sure that the referer is one of the key elements, as cutting and pasting the image URL into your browser's URL bar has - for all reports - made the image visible.
I'm also pretty sure that the browser string is key, since I have reports that the images are not visible from some browsers / devices but not others, even when the Coffee Shop is accessed in different ways from the same IP.
Solution - in "immediate mode", those of you who have the issue learn to live with it. I have ruled out going general image hosting on our servers (mainly because of the extra management software and support needs). If we move from http to https, it may provide a solution. And we could conceivable come up with a list of hot links servers that work for everyone. Longer term, a piece of clever coding could perhaps automatically mirror images, but all sorts of copyright issues there that I would need to think through.