Your chats are being saved on your phone and not on the Whatsapp server. The only moment Whatsapp saves your message is the moment you send it. The message is being saved on Whatsapp servers until it can be delivered to the recieving phone. This might take a while when that phone is out reach of internet or is turned off.
If the message is on the Whatsapp server for more than 30 days it will be deleted from the server.