To make an estimation about the odds of an encounter between two different planetary civilizations we will look to the history of our own planet as some kind of benchmark to help us to reflect on extraterrestrial biological evolutionary developments which might eventually lead to the emergence of extraterrestrial civilizations.
Maybe much to the disappointment of many of us: the odds might be very poor.
Likelihood that planetary civilizations emerge
The likelihood that the history of our Earth could have taken another turn in so many ways is very high. That could make us wonder about the likelihood that advanced civilization might emerge.
Further, the mass extinctions our Earth went through have shaped evolutionary developments and might even have been instrumental in boosting evolutionary developments. So does life on other worlds has to go through similar mass extinctions which eventually will lead to the emergence of advanced civilizations?
We should also keep in kind that it may be not that obvious that civilizations will develop technology which eventually enable them to reach beyond their planet. At our planet we can identify civilizations which are in equilibrium with their environment for thousands of years. And their conception of the world and their vision on life have proven sufficient to navigate efficiently through life and to find peace to build meaningful relationships.
Life on Earth has needed billions of years to reach a level that it became self aware and contemplative, capable of making a mental image of the universe and its constituents. But in only about 150 million years from now Earth will have become uninhabitable. That could bring us to think that on a cosmological time scale all these developments here on Earth are on a pretty tight time schedule. Maybe too tight to consider the emergence of advanced civilizations as a matter of course.
Lifetime of a planetary civilization
The estimated average lifetime of a technical advanced planetary civilization, capable of interstellar communication, of 10 million years might be an optimistic high estimate. We might consider ourselves fortunate if we make it throught the next 1000 years. If we consider the Egyptian civilization as the earliest one, starting more than 7000 years ago, then our civilization seems to have a total life time of less than 10 000 years, which makes the odds only worse.
(If we consider neolithic settlements as a type of civilization we might extend the lifetime of our planetary civilization with about 5000 years.)
Incredible high degree of synchronicity
The developments leading up to star and planet formation with a high metallicity (abundant heavy elements) and the evolutionary developments for life take billions of years. An incredible high degree of resulting synchronicity is required accross all those developments for having two planetary civilizations coexist in the same period of time, which might still be 10 thousands of light years away from each other. (For the sake of argument we will ignore relativistic effects which compel us to take more care when contemplating concepts regarding simultaneity.)
Speculations about very old advanced interstellar civilizations
One might speculate about extraterrestrial civilizations which have found ways to break free from their home planet and making themselves independent from its fate. Capabable to engineer themselves to adapt to various environments, to preserve genetic diversity and to spread out. Finding sustainable paths towards hopeful futures and finding ways to successfully avoid paths which will lead to destruction. These advanced civilizations might have become somewhat nomadic in nature on an interstellar scale and might have a very long lifespan.
One might wonder how such civilization might look like. A civilization, over one billion years old, which might have alienated from its orginal planet a long time ago, whose planet has become uninhabitable because its star is expanding. A civilization which has been able to harness huge powers which has enabled them to build an interstellar future. What kind of conditions, in the broadest sense, are required to have civilizations embark on such fantastic endeavors? Maybe such civilizations might be very alien to us. Our current state of humankind is even far from fit to build a very long term sustainable future on Earth. We might be even totally unperceptive for their ethical state of mind on a very fundamental level because it is inherent to our nature, not able to comprehend the required delicate fabric and the delicate state of such advanced civilizations.
It might be not that unthinkable that being focussed on an interstellar future is not something all living creatures derive purpose from. For example, for the vast majority of us it is fulfilling to find purpose in this life here on this beautiful Earth of which we are an integral part. Awareness about the existence of a large universe beyond Earth is sufficient for this majority. So civilizations which went interstellar might therefore also have resulted in new planetary civilizations from which traces to its origin have been wiped out deliberately at the initiation, resulting in successive civilizations which have been detached from its interstellar origin deliberately. Perhaps because the purpose of the connection has been found to become obsolete when sovereignty has been given, respected, cherished and protected.
On the other hand, our history shows that developments in science and technology are accelerated during wartime. The marvelous and inspiring Apollo program, for example, is unfortunately in fact a product of the Cold War. One might therefore wonder what kind of developments an interstellarfaring civilization had to go through. Such a civilization might have been forged by fire and rivalry and not through mental or sprititual development, so to speak. What kind of cement is used to provide the necessary cohesion assuming individual free will is still a factor in their social fabric? Caution might be advised in engaging in contacting such a civilization.
However speculations about very old interstellar civilizations still belongs to the realm of pure fantasy. It is very quiet outthere. Which could lead us to the conclusion that if life and the emergence of civilizations is likely to occur in this vast universe or even in our own Milky Way with its etimated number of 100 billion stars, the likelihood that they coexist is very small. On a cosmological time scale, in this endless sea of time, the lifetime of a civilization is likely to be just a blink of an eye.
Brief remarks about the origin of our Sun
When we look to the composition of our solar system, which includes our Sun, it contains heavy elements of which we have learned how they are formed during the life cycle phases of heavy stars. It has brought us to the conclusion that our Sun is at least a third generation star and very likely has been born out of the remains of multiple exploded stars like what is happening in the Eagle Nebula 7000 light years from here. It is therefore also likely that our Sun once was part of a group of stars but somehow got separated. It is assumed that this nebula, this cradle of our Sun, has been dispersed over the past 5 billion years. But the search to remnant of this cradle is a topic of research. To look for our Sun's siblings with their planetary systems might yield interesting results which might even touch us emotionally.