At least in Berlin, the influx of other european citizens did have a significant effect on the rents. I cba to find sources in English, but the city is trendy, and locals are annoyed by the masses of young hipsters who come live there, making rents rise, and don't even bother to learn german/integrate the slightest. There were for example articles about berliners who were flabberghasted that some bars and restaurants couldn't take orders in german.