The problem with that circuit is that it's going to splatter all over the shortwave band - an RF signal needs to be spectrally pure within legal limits, and within bands you are legally entitled to use.
A: You do not know what country the OP is and what the regulations there are if it is not the US
B: If run with a higher resistor and with a small antenna (FCC part 15 limits you to 10ft of antenna anyway) I do not think that it will cause any issues
I recommend using a resistor higher then 220 ohms to lower the output power
That circuit produces a square wave, not a sine wave.
A low pass filter can correct this but once again if you keep the output low (say 10mW), the chance of causing problems is very low.
You can get various legal and cheap modules for the unlicensed ISM bands, from simple wire-replacement tx/rx pairs to packet transceiver modules and all the way up to mesh network nodes like the Xbee's
XBees are neat since they work as a virtual serial port making them easy to use.
Be aware that if your circuit transmits over a sensitive frequency such as fire and rescue services or aircraft bands you run a risk of endagering other's life and safety.
This is extremely unlikely since those are in the VHF/UHF bands meaning it would be the 20th to 300th harmonics and they would likely be attenuated sufficiently to not interfere with multi-watt transmitted signals.
Also an effective aerial for 1MHz is awkwardly large anyway.
Obviously this is true. And also illegal to use an antenna of this size in the US on a part 15 transmitter.
Can I use the same circuitry as in the above link, and just change the code to "receiving code"? Im not sure what it would have to do, but i am imagining something like another interupt timer, also set to 1Mhz listening to data coming in on an analog pin - or maybe continuous listening, because the two devices might be completely out of sync.
This will almost certainly not works as you expect this is will pick up all kinds of electrical noise and crud. Without an input band pass filter, it would receive any radio signals from DC-0.5MHz. Even with a proper bandpass filter, sampling in that manner will not work well.
Also you would have to sample at 2MHz to receive at 1Mhz because the Nyquist frequency effect.
If communication between arduinos is your goal, then I suggest purchasing a module designed for doing this.