I read this rather depressing story about 10 years ago, I guess, in a collection that was probably much older.
The Project “Search for Extra-Terrestrial Intelligence” is to be terminated. It never found any proof of extra-terrestrial intelligence. The current Director, in charge of the termination, is throwing away everything in relation with the Project when he finds a file, containing recordings, that is not mentioned in the database. It was 20 or so years old, with the handwriting of the Director of that time. Alas, the latter has been dead for more than 10 years.
The current Director analyses the recordings in the file. They seem to contain indications that signals that were not just random noise were received from some star (I mean, a planet or any body in orbit around it; they certainly could not pinpoint it to the star itself). I forgot its name but as I explain below I just now "computed" that it had to be Procyon.
Since the Director at the time the signals were received was very enthusiastic about the SETI project, the current Director wonders why he did not make these results public.
Was he too careful because he was not quite sure and did not want to start a false hope ?
He starts searching the database for similar signals. He doesn't find any others from Procyon. But similar signals, though fewer and thus even less convincing, by themselves, were received a few years before from Sirius (that name I did remember - brightest star in the sky under our latitudes, after all ! ).
But even if by themselves the few, older, Sirius signals were not so convincing (and he did not fear to file them in the database at that time), put together with the slightly more recent and much more numerous ones from Procyon, it became obvious. That correlation was too much to be pure coincidence.
But then why keep the Procyon recordings in a secret file and not make them public ?
So the last Director discusses the matter with a close friend, a retired military officer who, IIRC, was Director of SETI before him, after the death of the Director at the time of the signals.
The delay between the arrivals of the signals, if there is any correlation between them, could come from the difference of distance travelled by light. Now Sirius is about 8.6 light-years away from Earth. The only bright star in the general directions of Sirius and just a few light-years further from Earth is Procyon (11.4 light-years away), hence my supposition. With an angular separation as seen from Earth of about 25°, I computed their distance to be just over 5 light-years away of each other. Rather close for two stars in an arm of our Galaxy. And this is quite consistent with the last few words of the story.
I don't remember the military man's reasoning, but he got to the conclusion that what was received by SETI was not aimed at Earth but that was just accidental eavesdropping on the military communications of Sirius and Procyon respectively. Further, he claims that they were at war at that time. And considering the interest of the first Director to history, (lots and lots of allusions to this interest) he is positive that the latter had come to the same conclusions.
So being highly advanced technologically, enough to have interstellar travel did not mean they had become more advanced in wisdom. War had not disappeared. Very depressing. Hence the secret kept about the discovery by SETI, in order not to spread panic.
Indeed, the two men are now really worried. With the distance of Sirius to Earth less than twice its distance to Procyon... If they are aware of us because of our EM communications....