Having lived in several different neighborhoods in my adult life, my answer is that it largely depends on the neighborhood. And, from my experience, socio-economic status is a big factor in the citizen-willingness to do anything on a volunteer basis.
Consider the fact that stay-at-home moms, for example, have a more directly vested interest and more time on their hands when it comes to community matters. People who are not parents, single mothers who have to work, or even married couples who both have careers, may not have the time nor the energy for volunteering in community programs, including crime-prevention.
On the other hand, citizens who have been directly affected by specific crimes, often hit the point where they will sacrifice in order to join a cause in preventing it further. In these kinds of cases, I think even the busiest of citizens create time to contribute to causes they feel are worthy.