The user who edit and save details of the repeating is the owner of that repeating. This user will be used in all interactions during execution of the repeated action, that means he will be:
creator of the issue, if needed by action (e.g. for the “Clone Issue” action),
editor of any fields of the issue, if needed by action,
the user who perform the workflow transition, if needed by action (“Issue Workflow” action),
In other words this user will perform any changes in Jira required by the definition of the repeating. This user can be (but not must be) the reporter or the assignee of newly created issue. Reporter and assignee can be set in proper fields in the form when setup the repeating.