Although I generally agree with the notion that the probability of two civilizations meeting each other is extremely low. I think there are factors that can improve it.

Even if a civilization only survives say 1 million years, it can definitely leave something that emits messages for a much longer period of time.

Now if we accept the previous point, and calculate the number of planets that are suitable for creating life, the probability might improve to an acceptable number. It's no longer just a few light years, but maybe a billion light years -- if we believe residues of a dead civilization may survive that long.