Connect two Arduinos via simple Serial connection
The RX/TX are the USB connection LEDs. They indicate whether your ATMEGA is transmitting (TX) or receiving data (RX) as opposed to the PC. If you're transmitting data, it's sent to both the USB converter chip and the connected node. After all, the USB chip is sending data to the host PC UART.

If you're receiving data from the connected node, the USB chip hasn't sent the data to the arduino, so no RX LED will be lit. It doesn't mean there is nothing sent though! The ATMEGA may still be receiving data, but the USB chip isn't involved, thus no LED will blink.

In other words, TX means data sent to USB, RX means data received from USB. These LEDs are directly tied to the USB UART converter chip, so you can't change their behaviour.

Having said that, my guess is that your code is probably working fine.
If you're unfamiliar to a piece of hardware, don't always trust it! Especially these things can be deceiving if you think that the RX/TX are related to your microcontroller, whereas they are only related to the USB UART chip. Solution: look up schematics, find datasheets and figure out when certain indicators are lit.