In Cassels's 1962 ICM paper (available here), he says the following: "Indeed, Tate and Šafarevič have, I believe, independently conjectured(5) that Ш itself is always finite", and the (5) is a footnote stating: "In his lecture, Tate denied paternity but adopted the conjecture. In conversation during the Congress Šafarevič expressed strong doubts."
So, maybe no one knows!
EDIT: As an added bonus, I found the following quote in Cassels's review of Silverman's book: "Without doubt the reviewer's most lasting contribution to the theory is the introduction of the Cyrillic letter Ш ("sha") to denote this group, a usage which has become universal."