The people in charge of the lab are surely responsible as well, but in the end people need to be reading MSDS's, do research on what they are using, and protecting themselves correctly, even if their PI doesn't say anything.
Yes, the responsibility is certainly a shared one. There is a general safety problem in universities, which is due to a large proportion of the workers being untrained. While safety may seem like common sense, I don't think it is really that simple. There are often still major problems with experienced people coming in from labs with poor safety practice - it is habit and attitude. For example, the detonation case study is particularly baffling to me - how you can decide to manually grind what you know is an explosive material without at least a blast shield is beyond me - but I was educated in a lab where we used blast shields for much less dangerous activities than that, so that's hardly surprising.
The major deficiency I've experienced in university labs is lack of enforcement. It's great to have safety lectures, safety policy, COSHH assessments for experiments etc., but when nobody is actually checking to see whether any of these regulations are being followed in reality (except for an annual inspection) standards remain fairly low. That said, I did work in one department where you were suspended from the for a week if caught without safety glasses in a lab. As you can imagine, this was an excellent strategy to get PIs to actually enforce good safety practice in their own groups - unfortunately I think that otherwise many PIs turn a blind eye to it.
The system we run in the lab I'm currently in involves a monthly safety rota, which gets everyone involved in safety and conscience of it. Since we started doing this about a year ago safety standards have improved substantially.