That won’t be possible, unless, say, you were to create subtasks assigned to each person.
A better approach is for these tasks to go to a project to which all candidate assignees are project members.
All Asana users should be managing their Asana Inbox, not just their My Tasks, and they are guaranteed to receive notification of the new task there (and optionally each may receive email notifications as well). Whoever wants to pick up the task should assign it to themselves in the right side of Asana Inbox and all others will see it’s already assigned in their Asana Inboxes and can just archive there.
Larry