Use Arduino as transmitter and receiver UART
You sure now Serial.write() and Serial.read(). That's already what you want. A hint to make it clear to you: When using the common Serial.print() function only the RxD line is in charge an thus only "half" of the serial interface is used. If you connect TxD of arduino A to RxD of another arduino B and RxD to TxD respectively you have full-duplex communication which is what you're asking.