What we do with EU migrants is up in the air, pending the outcome of Brexit talks. Asylum policy and non-EU immigration has always been our business, and not much has moved there, so the great and the good looked at the situation, found already that people had to meet X number of tough criteria and felt the added cost of the measures you're thinking of not worth it. As for right to privacy, I see no reason why it should be any different for migrants. For the time being at least, we already have access to national ID cards and health insurance information, shared intelligence and security databases re known criminals, terrorist groups, etc, and these have always been treated as special cases, mostly well, sometimes badly. To generalise from only the negative aspects of the system would set a bad precedent, cost money and without proper staffing and funds won't be that effective (generally systems which make unique exceptions all the time end up fragmented and inefficient and more open to abuse than simple, general ones extended from existing legislation).
