This is indeed how RSA works, and is seen as a weakness of RSA, among other things it can lead to accidental oracles if you use the same key to both sign and decrypt things.

To my knowledge, this is not how ECDSA works. ECDSA is "just" a signature algorithm, e.g. provides integrity and cannot encrypt or decrypt anything - in EC-land you would use a technique like ECDH to arrive at a symmetric key that you then use to do encryption.