Is it valid for a person to say "Ameen" to a supplication secretly in their heart, and is there anything in the Sunnah that necessitates saying "Ameen" aloud?

Saying "Ameen" aloud for a supplication is not valid except verbally. Intending it without uttering it is not sufficient, for the Messenger of Allah (peace and blessings be upon him) said: "When the Imam says, 'not of those against whom is wrath, nor of those who are astray,' then say 'Ameen,' for whoever's saying coincides with that of the angels, his past sins will be forgiven."