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.
Some of these mentalities can be shifted in a reasonable period of time, some cannot; again, there may be fundamental ways in which the ideals and actual culture of a place conflict that need addressing at a different level.
One simple tool I've found helpful at places where I've managed people and/or been responsible for developing the approach to work is having a clearly posted list of "odd jobs" that anyone and everyone is asked to look at from time to time, and especially if they feel they are running out of ways to be useful with their time. The list is especially helpful for people who have trouble internalizing the idea that they should work on things not assigned to them; it gives them "official permission" to step outside their normal duties.
What goes on the actual list? At my company, it was a mix of administrative and professional development tasks. Everything from housekeeping items (review and update our internal documentation, clean up the files on your computer, clean out your old email, look for spaces in the office that need organizing/rearranging, help get rid of old equipment, etc) to skill building (find a new resource to help with a skill you want to learn, watch an industry-specific webinar/video, learn a new tool and present it to the rest of the staff, find an ignored feature in a tool you already use, contribute a code patch or documentation update to some open source software, etc.) We tried to make the list broad enough that someone could find something that fit their mood, energy level and time available, but specific enough that anything they chose would benefit them and the company in some form.
Again, for some people these kinds of maintenance and development tasks are a natural part of the way they think about their role and how it connects to a larger vision, but for some people they may not be. If you use a list like this, find the right balance of offering suggestions for a range of personality types without being condescending.
I make sure to preface the list of odd jobs with some instructions:
Before starting on these odd jobs, please make sure you've done as much as possible to complete any tasks or projects assigned to you and any recurring responsibilities or commitments. Check with your team/supervisor/project manager to confirm that there's nothing else you could be doing for now as a part of the group's immediate goals. If you find that you are regularly filling time here with these kinds of odd jobs, meet with your supervisor or project manager to discuss that trend.
These instructions can help managers and supervisors who might be concerned about "releasing" staff members to spend time on these kinds of things; hopefully there's some sort of check-in before someone is off spending hours on housekeeping and skill building. But if in the end you find someone was quietly spending time on things that made your business/organization a more efficient/nicer/happier place to work, or that made them as volunteers/staff members more productive and knowledgable in their role, it's hard to see that as a problem.
Do you have an "odd jobs" list at your organization? What other methods have you found to be helpful for moving people with different kinds of working styles toward the ideal of "everyone's working on the right thing"?