1 expert and 1 parent have answered
If anyone can do anything with it, the original founders have to have the best chance.
Thing is, if you say Facebook and the other one...
Well at one time "the other one" may well have been Bebo, the perhaps not so long ago, Myspace.
These days, "the other one" has to be Google+
I don't compare twitter in the same light, as there is no real home, twitter and tweets are very much "in the moment", giving it more in common with a group chat system.
Bubbling under, there is also Diaspora, a work in progress, but listening to users, something which Facebook is VERY bad at.
The big problem for any establishing/rebuilding social network, is "but my friends are all on (name one)", so some USP is definitely needed, either by doing something so much better, or by becoming the place to be for a certain group of users.
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I think if the original owners manage to focus it solely on the young-teen school market as they so successfully used to then yes, they will manage it well provided they can make it "cool" like it used to be seen as.
Unfortunately they now have a huge battle against Facebook, Twitter and other social media that they didn't use to have. I think if they manage to get original content or features (like the equivalent of Orange Tuesdays, but specifically for Bebo users who are aged 10 to 15 for example) then they will do well.