Bas Yisroel — I haven’t learned all the Halachos recently, but my understanding is that the assumption is that information is private unless you know otherwise (i.e. if it was told in public it is obvious that it is not private). I vaguely remember learning some examples such as is a person tells you they are moving or switching jobs. Unless the person tells you it is public, or they say it in front of 3 people you should not repeat it.
commonsense — Your example is slightly different, as an engagement is something that is clearly public. (Assuming you were told they were engaged, and not just “going to get engaged”, in which case it is clearly NOT public!) It seems the one who told you made the error by giving the impression they were engaged when they weren’t formally engaged yet.
Bas, I do agree that you should do “damage control” to the best of your ability and try to prevent it from spreading further.