That's what they used to do in NZ, although you could leave it at the GP's surgery if you wanted to.
ETA: most people did, but I don't think they had any concerns about who would be looking at it (just the GPs really), and there wasn't the politicisation of personal details that you have in the UK now.
ETA: most people did, but I don't think they had any concerns about who would be looking at it (just the GPs really), and there wasn't the politicisation of personal details that you have in the UK now.