Wow, they haven't provided too much documentation for that tx/rx pair, have they?
Have you soldered the pads on the underside with the desired "address" code? It looks like these chips are meant to encode both address and data, but those specific boards are set up for hardwired address (via the solder pads) and then 4 data bits (D0-D3). For proper communication, the address pads need to match on the TX and RX side, and I suspect that it does not behave nicely if the pads are left unsoldered (floating).
I would try testing using a similar layout shown in the images that come with the "manual", using push-buttons on the TX side and LEDs on the RX side (no Arduino needed) to test to make sure that any signal is moving between the TX/RX paid before trying to send data across.
Cheers,