In the past there have been legislative efforts specifically to separate international policiing from domestic policing in the US.
The purpose of https://en.wikipedia.org/wiki/Posse_Comitatus_Act "is to limit the powers of the federal government in using federal military personnel to enforce domestic policies within the United States."
In the case of paris and belgium it would have been more helpful if the european agencies would work together. I can remember there were similar cases in the US.
One of those belgium bombers was supposed to be in prison, but the belgium police did not care. For the other one they did get a warning from the turkey, but it was ignored. Those two mistakes are only the tip of the iceberg.
I do not understand how throwing away our freedom would help in such cases.
Perhaps if we'd had better regulation while the CIA was running around destabilising parts of the world and installing the dictatorships that led to the current situation, we wouldn't be in this mess.
As it is, we know what it takes to prevent these attacks; a few generations of sane foreign policy that doesn't leave large chunks of various populations so full of anger and rage and humiliation that they're willing to blow themselves up just to hurt someone, anyone. Preventing an attack here and an attack there is just noise against the backdrop of civilian deaths every day from the groups we've created. So we could prevent an attack in Paris, at the cost of our liberty. A number of deaths that's basically noise. When do we prevent the everyday slaughter of people who aren't living in Europe at the hands of these same groups? How many of those deaths are prevented by me surrendering my privacy? Why should we give up our freedoms to prevent a tiny handful of deaths against a backdrop of daily slaughter? And furthermore, give them up while I can see our leaders, instead of beginning the three generations of sane foreign policy, continuing to exacerbate the problem? Maybe if the deal was that it was a short term thing, just while the root of the problem was fixed, I could get on board, but I see no sign of that.
Frankly, giving up our freedoms is the pathetic, easy choice. It's the cowardly choice and the way to refuse to deal with the real issues, because the real issues are expensive and hard and will take a long time.