I'm a grad student in California (but not in the UC system). Just want to add that I know the departments at the UC schools have individual budgets as well, so if you are at, say, UCLA, you are being paid by UCLA to do work with them, not to work for another school. That is, I would say there is as much of a connection in the sense you're asking between UCLA and UC Irvine as there are between UCLA and University of Washington (i.e. none).
And, unlike undergraduate admissions to the UC system, there is no single centralized UC grad admission system. You apply to each graduate program separately. In fact, they don't even use the same application software (when I applied, UCSC used "embark" and UC Berkeley had its own customized system).
In essence, for the purposes of the question you're asking, I would recommend that you treat each individual UC school as completely separate schools that just happened to begin with "UC". Of course, collaborations across UC campuses are still possible, just like you might expect to find collaborations between researchers at MIT and Harvard (both being in Boston area). However, this is less likely due to the fact that they are "UC" schools and more likely due to the fact that they are in close proximity to each other. As cwr pointed out, there is a nice cluster of UC schools around UCLA area (don't forget UCSB too!) and there are often cross-campus events. For example, there was a recent mini-conference on exoplanets for southern California, and we had people from all of these UC schools, as well as other schools in the area, like Caltech and CSU Northridge. In other fields, USC is another good school in the area that could participate.