Maybe there aren't too many people that would use this, but I sure could. When you look at the RSS feed of someone on your friends list, if you're logged in, you see their Friends-Only posts as well as their regular Public posts. However, most RSS readers aren't ever going to be capable of simulating a login to LiveJournal in order to have the proper cookies set in the proper places so that these entries can be seen in the feeds. Therefore, what would work better is if, at least the RSS page, supported HTTP Authentication. Then, a URL like http://myljusername:firstname.lastname@example.org/users/data/rss would allow friends-only posts to be seen as well. If the developers would prefer to keep the LJ password from travelling the network in plain text, perhaps a PIN could be entered somewhere to serve this, and only this, purpose. If none of that is possible or desired, then perhaps the page could simply return an RSS item containing the words "private/friends-only entry" with a link to the entry so that at least notification will be received.
My homegrown RSS reader currently detects LJ requests and sends my LJ cookies along with it so I can get all the information I desire. However, I don't think there is another RSS reader that would.