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So, I would assume that you would be communicating through radio waves. If an object broadcasting a signal moving at the speed light away from you, I would further assume you experience a severe Doppler effect. To the point that I don't think you would experience anything coherent. You would receive small packets of information at a time, scattered across several million years.
This is just my initial impression on the fly, do not take this as any sort of gospel. I also did some communcations work for a time. So, this is tickling my brain and I might spend the rest of my evening in my books.
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