Is this a sensible scheme for 2-way client/server communication? A client connects to a server. The server and client both generate RSA keys, and send over the public ones to each other. If the client wants to send a message, they encrypt it with the server's public key, send it over, and the server decrypts it. Vice versa with server -> client.
Does this make sense? Presumably breaking this is equivalent to breaking RSA, right? Is something like this ever used in practice? If not, what do alternatives do better?