I can’t say I know the definitive answer to your question, but I do know from personal experience that you pay tax in the country you are considered to be resident in so PAYE for UK tax would not be applicable.
You don’t have to be self-employed in this situation and arguably the person working for you may lose some significant benefits if they do become self-employed.
I don’t know how it works in Malta, but in the UK technically HMRC determines your relationship with the organisation you are working for. So while you may set yourself up as self-employed if they decide to look in to your tax arrangements and decide your relationship with the person paying you is more like employer and employee (and there are criteria for this) then they will tax you according to the status they give you. Whether that is similar in Malta I don’t know.
It might be possible for the person you are talking about to run a payroll system in Malta for paying tax there as an employee and you would have to comply with the law there in terms of employment rights.
I’m happy to be corrected, but that was the information I was given when my relationship with a Norwegian company was discussed with HMRC.