No, not automatically. You can always make copies manually. Bu why would you want to make a copy?
I have ATSC OTA tuner cards in a spare PC located right inside the same wall that has the outdoor antenna.
Some months back, I tried various TV software, and I ended up with "Argus TV" as the best recording software (very stable and very good feature set). Unfortunately none of the open source playback programs work very well (XBMC, Media Portal) and in fact, JRiver MC19 works best to play back the completed TS format recordings.
The wireless bandwidth to one room in the house is not enough for real-time playback of ATSC 1080i recordings, and the person in that room wants to do so. (I've tried repeaters, powerline, etc ... only an ethernet wire through the attic would work, and that's a big project.)
Argus TV allows one to run any command upon completion of a recording, and the filename of that recording is a built-in variable, so a one-line batch file copies the file to the laptop in that room. Whenever it is complete, then the person can watch.
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Recently, there has been so much interesting development of the TV software in MC19 that I was hoping to give it another try, but the above is one of the major uses of the OTA tuners in our house.