Redmine

General permissions can generally not be applied selectively based on issues attributes such as author or assignee currently. As such, the permission to delete issues can only applied to all issues visible to the user in a project or none by setting the generic delete issues permission in the user's role in the project.

In general, I would recommend to only grant this permission very sparingly to avoid accidental deletion (which is always final and can not be reverted). Instead, you should define a suitable status workflow to close or reject issues instead of deleting them., That way, closed issues can be checked later and re-opened if required and you don't risk losing important data by accident or due to malicious users. By default, Redmine shows only open issues in the issues lists but you can define a custom query if desired to select the displayed issues.

Redmine can deal with a lot of issues efficiently so there is generally no need to delete issues for performance or "cleanliness" reasons.