This is by design, in fact—because Trello Power-Ups are only connectors between multiple services, they don't expose any data that a person doesn't otherwise have access to. In your scenario, someone does need to have a Jira account in order to view that content.
One option that you might want to look at is using Unito's sync feature: https://info.trello.com/power-ups/unito
Doing that will sync content from Jira over to a Trello card, which wouldn't rely on the other person having a Jira account to view the content!
Instead of Power-Ups, you could try an app for Jira that can do the sync in any direction you'd like to - from Jira to Trello, from Trello to Jira or both. This is very flexible, allows to sync not only statuses, but whole issues! And no one is to worry about another account in a program that one doesn't use :)
Check out the webinar we hosted to know more. However, keep in mind that this app is only compatible with Jira Server, so if you're on Cloud, it won't work.
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