The Snowden files showed in great details how European countries used US intelligence to spy on EU citizens and then request that data through intel sharing, thus bypassing local and EU law. It was an effective way to get the benefits of spying on your citizens with a plausible deniability that it was the Americans who did it, and that the fact that data was shared is simply a fact of Nato and other deals between EU and US.

Obviously this is not something EU citizens want. If we wanted it, we would issue laws that gave the military and police the right to do it themselves. The only reason that this roundabout way came to exist is that such surveillance would not pass unnoticed by voters.

There are some "more local" alternatives. Sweden for example can (and as rumors goes, do) use neighboring countries like Denmark to spy on Swedish citizens by looking at network traffic that goes over the border. People have argued however that this is a bit worse of an deal since you don't get access to the larger intelligence network that US has, and you also have to trust your neighbors with possible sensitive data.