Would it be possible to use to of these on two separate arduino's, to send and receive information? I am having difficulties understanding the chinese-english descriptions and haven't seen any documentation on doing this. There would be no phone involved. There would only be two arduino's with a Bluetooth module on each of them sending and receiving data from each other.
The boards in your link look funny. Typical boards like this include the "bluetooth circuit" which piggy-backs onto that board, and is soldered at all those little pads.
Have a look at say, the HC-05 board ~ https://www.google.com.au/search?q=hc05+bluetooth+module&tbm=isch
It's easy to see the part that's missing from the one in your link.
For these HC-05 boards, it's possible to pair two together. They need to be configured as "Master" and "Slave" though. Some bluetooth boards can do slave-mode only, so be careful before you purchase.