Sort of. If you define "their issues" as "this person is the reporter", you can use "reporter" in the permission scheme. Same for "assignee". You could also define an "owner" custom field and use that.
However, before @Joe Pitt gets here, I'd like to strongly recommend not allowing anyone to delete. Issues are, well, deleted. You can't get them back or report on them, so if someone is a bit too keen, you're looking at having to go back to backup tapes if they kill the wrong thing.
There's other tricks.
A good one is a "wastebin" project that you move the issues into instead of deleting. It's a bit of a fiddle for the users, so it encourages them not to raise junk issues.
Another friendlier one is to a "redundent" status in your workflow with a "delete" transition going into it. On it's own, that makes it clear the issue is to be ignored so you can use filters and boards with "... and status != redundent", and for a bit more automation, also play tricks like set a security level or workflow property that hides things in that status.
thanks this is great feedback! I am trying to define and recommend some best practices, although it is not my first turn as a admin, I am new to Jira.
I like the redundant option, so they can see for themselves how many that are actually creating in "error". I think that part of the problem is that the users may be using our prod environment to test. (I know, we do have a test environment - smh) Anyway, so they want to be able to delete.
Watsebin - will fill up fast. I worry about the unnecessary load on the system.
The wastebin isn't more load than a "redundant" status really, but I prefer the status because you can instantly see which projects are creating lots of junk.
For both methods, an admin filtering for "stuff to delete over 1 month old" and using their bulk-edit rights to delete is a bit of a chore, if you decide you do want to clean it all out regularly, but less of a chore than restoring from a backup when someone messes up.
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