Judging by my own experience and observations, there is a subtle inter-generational shift in cultural identity. Only by the third generation do people feel completely assimilated into the local culture. The immigrants never quite "make-it." I think of myself very much as an American. But I am frequently conscious that my accent and my experience causes me to stand out a bit - not in a bad way, but it is noticeable. Many of my fellow immigrants feel this to a greater extent. It makes for cultural tensions, but also for an exciting dynamism, an infusion of cultural DNA into to social gene pool. My children are American. But even they, as children of South African immigrants, carry an extra bit of background baggage. They feel just a little bit different, in possession of something unique (the tacit and conscious knowledge of another culture). You see this most clearly in the way they gravitate to other first generation Americans in their social circles. By the third generation the children are full Americans. Their grandparents might as well be from another planet.
This pattern is repeated over again in each generation of America's population. It has been a necessary part of the ongoing vitality of the American experience, including its entrepreneurial component. Without it American exceptionalism, and the rate of value-creation, of economic growth, will not be sustained.
Illegal immigration is not a problem. It is an opportunity.