Google Plus seems to have achieved this fairly well. I haven't had the time to take a look at their API yet, but I can't imagine it's any less than what LJ provided.
Here's another one: Dreamwidth runs an LJ-derived codebase, arguably an improved one (they had considerably better separation of “subscribe” and “authorize” last I checked, rather than a “friends list” that conflates these), and some of the people I contact on LJ have moved there, but they all have continuous crossposts back to the original LiveJournal, and if I moved there I think either no one would read anything I wrote, or else the comment streams would be so disjoined that I would be effectively a strange-looking LJ user anyway.
That last is also a concrete example of why nonexclusivity is not a complete solution. The resource that's being fought over is not where one can read but where a bunch of other people do read. If everyone views your content at Phuubaar's House of Crossposts, then if Phuubaar cuts you off, you are still hosed in the general case even if you provide the same stream somewhere else, because those users are not going to know about it or are going to find it too inconvenient to subscribe.
And the tooling around Atom and RSS aggregation all seems to be built around the idea that feeds are almost always public. I haven't had any success with the idea of creating a private Atom feed and expecting any of my friends to be able to read it. Either it'll require authentication, at which point the software usually won't be able to access it, or I can try to use a capability-URL style, at which point one of them will punch it into their favorite everyone-shares-everything social aggregator (Google Reader?) and then my (illusion of) confidentiality is gone.
This is terrible, and I don't know how to fix it.