Hazardous trees are a difficult subject to explain when talking to people. Trees that are diseased or structurally sound can be very dangerous to life and property, in fact that what make them hazardous; having a 'target' of property or any area frequented by people. And as has been stated before, when a tree that is hazardous that has been identified as such by a professional there is a clear responsibility to do something about it. The law is not on the side of anyone that ignores the existence of a hazardous tree should it fall and injure or kill someone. I have seen many trees though that I would deem potentially hazardous that stand for many years. That being said, it is impossible to say with 100% certainty if and when a tree will fall. Erring on the side of caution is probably the best policy. Especially if you have an obligation to the public's safety as most municipal arborists have.
Some members of the public, and I'm not saying anyone on this thread does, don't see trees as living things with a limited lifespan. They, in the case of large trees especially, have 'always' been there. The fact that a mature tree may be at the end of it's lifespan and under stress from competition or from environmental factors could be susceptible to diseases or insects is not known to them. It's a matter of continually educating people and helping them understand trees don't live forever.