"Is this person a serious danger for himself/herself or other people?"
Note that it's not the institution the one to make the decision, but a judge. And, also, taking completely someones freedom away is a very extreme measure that will only be considered for really extreme cases, there are a lot of middle grounds, from compulsory regular visits to the doctor to home visits from the doctor. But all of this, is not usually necessary because most of the people that are mentally ill, know that there is something wrong with them and will voluntarily seek treatment.
It's at least peculiar than when people from the US are asked about mental healthcare, their visceral reaction is "OMG!!! THEY ARE GOING TO TAKE AWAY MY FREEDOM!!!". So much fear, so little idea about how an actual mental care system works.
Source: exGF psychiatrist in a western European country with a strong public healthcare network.