I've never thought about it for myself to be honest. Where I live I can log into a central website called 1177 (which is also the phone number you can call to talk to a nurse over the phone for advice if needed) and see all of my visits, doctor's notes, and test results from different clinics. Maybe if I had a more extensive medical history and had to visit more obscure clinics etc I'd request printouts of my test results and such.

I have thought about this for my cat, however. The vet clinics here also regularly transfer journals to each other as needed, but I don't have access online the way I do for my own journals. Unlike myself (knock wood) my cat has had extensive medical issues as a kitten and has a very long set of journals from different locations. Every once in a while I ask the current hospital we frequent to send me a copy of his journals. Sometimes they give me physical copies and other times they send them via email, and I be sure to have these on-hand just in case I need to provide them in a rush to an emergency clinic.