Within any given business or organization, the ideal is (probably) that everyone working or volunteering there will not only take on all of the tasks and projects that are clearly a part of their recognized role, but that they'll also work on things not necessarily assigned to them but that are still useful to the overall goals of that organization.
If everyone shares the same vision and goals, and everyone participating is sufficiently empowered and inspired to work toward those goals, this ideal can be easy to realize. There is almost certainly always something else that can be done to support or further the mission of the places we work and volunteer; if there's someone who regularly ends up having some "down time" with "nothing to do" then there's probably something else wrong - with the person's mindset, the structure they are working within, or the organizational culture overall.
In practice, I've found that people have different personalities and personal/work backgrounds that lead them to respond differently to this idea of "working on things that weren't necessarily assigned to me." For some, it's a no-brainer and they can jump right into that mentality. For others, it represents a threshold of riskiness and potential for failure that they may not be willing to cross: "If it wasn't assigned to me and I do it wrong, I don't want to be responsible for the outcome." For still others, it can just be the challenge of imagining tasks or projects outside their job description or previously assigned duties; the inertia of working within familiar problem spaces is hard to overcome.