Well, the terrorists, murderers and serial killers are being searched, arrested (when cought) , put into trial sentenced and put to jail.
But they are not put to death (at least if during their arrest, they don't force the police to defend themselves).
About Europe understanding the US, it's mainly in the international agreements where there are changes. After all, all those treaties have purposes, reasons for existing and even if they often only reach a low common denominator, they can serve as a basis towards something bigger.
The EU's origin can be found in the Coal and Steel Community way back in 1951.
I guess the problem is knowing and accepting to delegate power. When a country becomes federal after having been centralized, there are many issues that need to be delegated to more regional authorities. On the other hand, there are issues like pollution, human rights, international conflicts, terrorism that are much bigger than one country's borders.

Raoul