Forward secrecy normally requires a handshake where both endpoints are online. Signal is obsessively forward secret. The Signal Protocol's claim to fame is that

Forward secrecy normally requires a handshake where both endpoints are online. Signal is obsessively forward secret. The Signal Protocol's claim to fame is that it can do offline messaging while still providing forward secrecy. It very much requires a server to store the required cryptographic state to do so. I guess the restriction could be considered some sort of downside that comes with protocol complexity.