When the European Union’s justice commissioner traveled to California to meet with Google and Facebook last fall, she was expecting to get an earful from executives worried about the Continent’s sweeping new privacy law.

Instead, she realized they already had the situation under control. “They were more relaxed, and I became more nervous,” said the EU official, Věra Jourová. “They have the money, an army of lawyers, an army of technicians and so on.”

Brussels...