Figures I've seen state 20% of Irish population of 5 million were born outside of Ireland. Given the diaspora, many of those have Irish parents. 300,000 of that million were born in UK. Suspect similar from US, Australia etc. 100,000 from Ukraine. Not sure where couple of million comes from, but...
Can't speak for all Europe, in Ireland I'd say a significant lack of building, greedy buy to let landlords/holiday home owners and poor infrastructure outside of Dublin