The people in the major "settlements?" They are Israelis. They vote in Israeli elections, follow Israeli law (to the extent that any Israeli bothers to follow laws that they disagree with), and are Israeli citizens. The major cities around Jerusalem are Israeli in everything but name. Outside those major blocks, things get more complicated.
There are settler groups that are rabidly religious and loathe the secular Israeli government. There are hilltop youth who are religious radicals who have no loyalty to any government, and only loosely follow a handful of radical rabbis. There are Israelis in Hebron which are Israeli citizens and who are loyal to Israel, and are upset that Israel is not supporting them more while they try to live in the second holiest city in Judaism (whist both Jews and Arabs in the city constantly do their best to make the other's life miserable).
There are Arabs living in the major blocks who are loyal to the idea of Palestine, but have no desire to live under PA rule. There are other Arabs who live under Israeli law, but have PA citizenship and are loyal to the PA (at least depending on who is asking the questions). Go through the area and the political and cultural landscape can change dramatically from village to village, town to town.
As far as cultural integration goes, I've not spent a whole lot of time there, nor have I done much more than speak to a few of locals, so take the following with a grain of salt. My understanding is that while there are economic links between Jewish and Arab communities in Area C, and to a lesser degree area B, there is very little cultural cross pollination, at least in the smaller villages and towns. In the cities, Arabs and Jews work together, and in many cases live together (though there are many neighborhoods known for being 'Arab' or 'Orthodox' many more are mixed). In the smaller villages and towns in the West Bank though, you have Arabs who will travel to Jewish towns every day for work (or more rarely vice versa) but they will rarely live in the same towns or villages (though of course there are well known exceptions). Each population tends to speak different languages, get educated in different schools, and are generally separate socially and culturally.
In Israel proper there is more effort made towards integration. Within areas B&C Jews and Arabs may be dependent upon each other economically, but there is a clear divide based on nationalism and the laws each lives under. Most Arabs are citizens of the PA (and thus must follow PA laws) while all Jews are Israeli citizens and thus are usually (though not always) subject to Israeli civil law (on occasion they are subject to Israeli military law instead, as are Arabs living in Area B). Where things get really annoyingly complicated is when you have issues where Israeli and PA citizens are working together in places where its not clear if PA or Israeli law should apply, and so you have Israelis being punished for breaking Israeli law while following PA law, and Arabs being punished for breaking PA laws while following Israeli laws. It gets really messy. It gets even messier when you take into account that there are businesses (both Israeli and Palestinian) which set up in these areas specifically to take advantage of the legal grey zones which exist especially in Area B.