They firmly state you must have the appropriate rights for sharing, but they trust on your say-so.
Every story sharing site has to take their poster's work for having the rights to post a story. Absent mandatory registration of copyrights, there is simply no way for anyone to verify such claims up front.
At least one of them seems to be ripped from Project Gutenberg and all passages about Gutenberg removed. There is one statement about the expired copyright with the same wording as in Gutenberg books.
And that's perfectly legal, at least in the US.
All the stories on project Gutenberg are public domain (expired copyright).
And as to the statement about expired copyright:
Under US copyright law, simple facts, (name/phone# data, sports scores, that a book is off copyright) can not be copyrighted or protected as a subset of a larger work.
There are too few ways to make the expired copyright statement in a grammatically valid way. Any attempt by Project Gutenberg to try and claim copyright on that statement is US courts would be doomed to failure.