Normally, the owner of the Jira issue should be the person working on the issue. The owner of the issue could be changed (reassigned) if one is making modifications to an existing issue. However, although this is how we have normally seen this performed, it is up to the organization to determine who needs to be the issue owner based on their internal procedures, best practices, and regulations. 

For example, IEC 62304 states

5.6.7 Integration test record contents

The MANUFACTURER shall:

a) document the test result (pass/fail and a list of ANOMALIES);

b) retain sufficient records to permit the test to be repeated; and

c) identify the tester.

In this case, you may want to ensure that the owner of the Jira issue is the person performing the Test Case configuration item in order to meet the needs of this clause.