Given a message that is encrypted using a Trivium and the key that was used to encrypt it, how does the receiver decrypt the message?
Trivium is a synchronous stream cipher. For synchronous stream ciphers the key stream is independent of the plaintext and ciphertext.
The key stream (which is not the key itself, but the output of the cipher construction) is then combined in a reversible way with the plaintext to for the ciphertext (commonly byte-by-byte). The reverse operation with the same key stream and the ciphertext brings back the plaintext.
Usually XOR is used, which allows you to use the exact same construction for both encryption and decryption (as $PT \oplus KS = CT$ and therefore $CT \oplus KS = PT$).