[alibarber]

It varies. Ironically the UK never seemed to bother with any formal registration[1], apart from getting a NI number for work (like everyone has to do).

I couldn’t register in Finland as an EU citizen (when I was one) without an employment contract or proof of self sufficiency. You have three months from arriving to register. It's true that if you don’t you're unlikely to be actively tracked down - but good luck trying to achieve anything here without that registration.

[1] I.e. with a government body responsible for enforcing immigration. It is likely that you would be asked to register to be on the electoral register at the house you are living in. This is not really the equivalent, I am talking about going to the Immigration authority of the country with your passport and papers.