Re: Cat ... bag ...
If you've got one foot in each country, e.g. income from the UK, living in another country then yes, you need two tax returns. If someone has no taxable income in the UK and is living and working in the other country, there's no need to complete UK tax returns.
It is obvious if you're working in another country you declare residency, pay tax and social security to that country, and use that country's social security system and health service.
What freedom of movement does is give you the right to be a resident, compete on an equal footing for work (no need for companies to sponsor you or prove that nobody else can do the job) and be treated like any other citizen of that country when it comes to the welfare state or healthcare. You may have to do a bit of paperwork to achieve that, but it's not any more onerous than a citizen of that country would have to do either.
And now after Brexit that is not true any more. The loss of FoM amongst many other things makes Brexit the greatest swindle perpetrated by a country's government against its own citizens in recent history.