Dany's perspective of her father

[Rhaenys_Targaryen]

Stannis was sent to Dragonstone to capture the Queen and Viserys but they were fled before that. It is common knowledge that Dany was born in Dragonstone amidst a heavy storm. Even Kevan Lannister calls her 'Daenerys Stormborn' (DwD epilogue). I don't know how everyone knows that from White Harbor to Old town, but they know that Rhaella gave birth to a daughter named Daenerys before Stannis stormed Dragonstone. And it wasn't Viserys telling them.

Only Darry, four guards, a wetnurse, and Dany and Viserys escaped Dragonstone. All other people, who had been present for the birth, and who had been told about it, were captured by/surrendered to Stannis when he took Dragonstone. That's how Westeros learned of it all.