There's not really that many people to spread the rumours.
The Battle at Falme had three actors:
- The Whitecloaks -- who were all but destroyed. Their leader had set an officer to observe, but he became obsessed with pursuing Perrin rather than spreading the truth of events.
- The Seanchan -- who aren't the most talkative lot when around normal people. They also retreated across the seas to regroup.
- The Army of Shienar -- recently having lost their commander -- who are loyal to the new Dragon (mostly) but then come under the thrall of the Prophet.
None of these groups are particularly trustworthy for a lot of the series. For a long time, people don't believe the Seanchan exist. People ignore the Whitecloaks wherever they can. And the Shienarans go hide in some hills very shortly after the battle.
There are also other events going on -- most interestingly the battle between Rand and Ishamael in the sky above the battle. This scene was more easily observable for people not directly involved in the battle, and so rumours of it spread more quickly.
No one announces that the Horn was used. People don't make the connection between a legend no one really believes in and a battle people think is just propaganda for another false Dragon. By the time Rand is more accepted, there's other things to talk about and the Horn is safely locked away in Tar Valon.