You can transfer using hardware serial (pins 0 and 1) which is more reliable than software serial. The fact that the Tx light doesn't blink is irrelevant.
TX pin won't blink (transfer data)
Pins don't blink for a start, and whether or not an LED blinks doesn't necessarily indicate if it is transferring data. For one thing, at high baud rates, and small amounts of data, the blinking would be hard to see.