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The way the audio driver works is that both CDJs are aggreagated into one "device" (as seen by the software). When one of the hardware players is either disconnected, or switched from PC mode to USB / CD, it will remove itself from the driver. How the software reacts depends on how it's designed / configured for monitoring the devices. I know Virtual DJ, and rekordbox don't like it when audio devices are removed from an aggregate. This is not a bug, but rather a recommendation that you don't unplug one of the CDJs or switch its mode while in use.
In an upcoming version of rekordbox, you will be able to simply connect your friend's USB drive to your computer and their export device will be visible within rekordbox DJ to perform directly from that device as well as your collection.
Sorry this isn't the answer you're looking for, but the technical complexity of not having the device disconnect is quite difficult to sort.