Traveller,
I don't know 'for sure' but my HP fax machine has a memory function that supports an out of paper condition. But once the paper is replaced and the fax printed, the data or document is no longer in the memory.
I would guess the data is stored in the computer's print function until its printed and then gone. I doubt it is in the printer memory. If you've had occasion to look at the Spooler function, the process that lines up print jobs when the printer is busy or unavailable, you'd see a list of things waiting for the printer, but not transferred to the printer.
I think you're safe.
Bill
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