The problem here is one of privacy. RSS, by convention, does not typically involve any authentication. So there is nothing to keep me from subscribing to your global inbox feed, if it were implemented standard-like.
HTTP authentication could be applied, but that would break RSS feed readers' ability to read the feed.
Another option would be for each user's feed to have a unique, hard-to-guess url, but that still fails the security test in my mind; unlike typical authentication, where you can change your login info if it is compromised, such a system would not necessarily.
Of course, it could further allow regeneration of each user's feed url but... now we are getting into complex setting territory. Also,changing your feed url would 'reset' the feed in your readers (it would literally be a different feed), so it just seems to go against the middle "S" of RSS.
I think the easiest opyion would be some sort of desktop notifier, of which I think there are some on Stack Apps. I also seem to recall something that lets you feedify your inbox even, iirc... which renders my whole "stay off my lawn" argument somewhat moot, if true!