In JIRA it is completely impossible. A task has one project and one status. You're lucky to have view and comment rights on another team's task, forget about change-status or reassign. An admin can "move" an issue but then it's gone from your own world.
The net result: the issue tracker is no longer a communications medium, but a paper trail for the bean-counters, where you laboriously log conversations and decisions that actually occurred by email or Slack. There will be a JIRA corresponding to any give issue, but reading the JIRA will not tell you the story anymore.