The latter, apparently - although nothing was stated out loud.
Well it must have gotten out somehow; how else would Picard have heard about it?
About what? Picard in the final timeline never heard about any survivors to the E-C debacle. All he knew was that the ship had been lost fighting the Romulans - a bit of information that would easily have reached him because the loss had been at a Klingon colony, and Klingons were allies. Perhaps Romulans in that battle conquered and annexed Khitomer, making it difficult to get information out later on, but we never quite learn this, and it wouldn't affect matters much anyway.