How did Albania manage to lose such massive amounts of its population without even being an EU member?

It looks like some commentors misunderstood your question and thought you were asking why so many Albanians migrated out of the country, not *how* so many of them migrated, i.e. how did so many of them visas and residencies. I think the answer is similar for all non-EU Balkan countries. After the Second World War West Germany underwent a period of economic revival called Wirtschaftswunder. Germany needed a lot of workers to rebuild itself and to satisfy the demand for labor they invited guest workers from Southeast Europe to fulfill that. In fact many countries in Western Europe had their first wave of foreign immigration in this period - Britain received its first immigrants from the West Indies, France from the Maghreb and West Africa, Germany from the Balkans, etc. Those workers eventually made connections with each other and established themselves in their host countries. The heyday of the Balkan criminal underworld was the 70s and 80s. Arkan, a notorious Serbian gangster, was known to have committed petty crimes with his gang all over Western Europe at this time. Then came the Yugoslav Wars in the 90s when people massively applied for asylum. By the 2000s a lot of Balkan people already had hundreds of thousands of family members and businesses abroad who could sponsor them. That's actually the most common way to immigrate to America - having a close family member who sponsors you. It's a domino effect. Albanians have big families so they're able to immigrate in large numbers because of that. There's also special programs like the Westbalkanregelung that makes it easier for people from the West Balkans to find a job in Germany.