To move tickets back and forth and have requests go into JIRA, I recommend you to try Customer Case. It fits perfectly to your purposes.
With this add-on you can create feedback/support forums based on your JIRA Cloud and then store users' ideas as JIRA tickets.
Try it. I think it helps you a lot.
you have to write a custom add-on for such integration. you can try to build it yourself or hire an Atlassian Expert company.
Here is how you start:
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