In the UK, internships are open to all residents of the country, but you do have to have been living in the country for 6 or 24 months beforehand depending on the internship you want to do. The good news is that a substantial number of interns end up with a job offer. 

Where I live in Spain, you have to be registered with socia welfare, and accepted into a job placement program or "Taller de Empleo" to qualify, but there seem to be very few internships and very few that lead to job offers after they've finished. 

allowing only nationals to do internships would be an obvious discrimination.

But still, it is so damn hard to get an internship here !