That’s a terrible idea. It means that anyone on the list–or with access to it–can hijack or break your site (perhaps without meaning to) without knowing who it was. You should give everyone their own admin account.
But that aside, I only ever remember seeing a single notification per site. You can look at /admin/email/sent and see if it’s listed there multiple times. I don’t think Discourse is doing it, but I don’t know why your mailing list would either.