Should I write "we" or "I" in my research statement?
A research statement is a mix of past and future.  

When you are talking about the past, you should be honest about the fact that you are not working in isolation---in fact, that is a good thing.  Use "we" or "my collaborators and I" or whatever most accurately describes what actually happened.

The other key part of a research statement, however, is your views about the future: what you think is important, what your ambitions are for the future, etc.  Those are unambiguously your own opinions, and should be "I" statements.