The operating paradigm here is one task - one owner. (Although you can obviously assign nmultiple resources to a task.)
If checklists do not work for you, you could try working with subtasks. Alternatively you might have to define a operational guideline, defining who may set the task to complete.
Another alternative might be to define generic resources that designate your teams and initially assign tasks to these generic resources. When a team member decides to take ownership of the task he or she can the re-allocate the task to him or herself.
