Has a reference manager ever accidentally shared unpublished data?
This has happened deliberately, where an informal reviewer made the work of an author public without their authorization. One well-known example is Wallace's twenty-page abstract of what would become known as the theory of Natural Selection, which was mailed to Darwin on June 18, 1858 with a note asking it to be forwarded to Charles Lyell. Instead, Darwin, Lyell and Hooker decided to publish Wallace and Darwin simultaneously so that Darwin wouldn't lose credit for the work he had already put in (and would put into The Origin of Species and other books). This paper was read out to the Linnean Society of London on July 1, not leaving enough time for a letter to be sent back to Wallace in Malaya to check if this was okay. It's pretty clear that Wallace did not intend this manuscript to be published (as per Prof. Charles H. Smith):


  For his own part, Wallace later indicated in print on no fewer than four separate occasions that the manuscript he had sent to Darwin had not been intended as finished product: in an 1869 letter to the German biologist Adolf Bernhard Meyer (later reprinted in 1895 in Nature, Volume 52, on page 415), as a note added to the essay when it was reprinted in 1891 in Wallace's Natural Selection and Tropical Nature (on page 27), in the article 'The Dawn of a Great Discovery' in January 1903 (Black and White, Volume 25, on page 78), and in his autobiography My Life in 1905 (Volume 1, on page 363).