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 that Earth has experienced have shaped evolutionary developments and may even have been instrumental in accelerating evolutionary developments. So the question is whether life on other worlds has to go through similar mass extinctions which eventually will lead to the emergence of advanced civilizations?
For a more detailed look at the geologic time scale: Geologic Time Scale
We should also keep in mind that it may not bet that obvious that civilizations will develop technology which eventually enables them to reach beyond their planet. On our planet we can identify civilizations which hve been 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 where 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 technically 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 through 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 lifetime 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
It takes billions of years to have a planet on which life has emerged. First, an evolutionary process is needed during which a star and planets are formed with a high metallicity (abundance of heavy elements). And subsequently an evolutionary process for life to emerge. The likelihood of having two planetary civilizations coexist in the same period of time in our Galaxy, which might still be 10 thousand light years away from each other, is extremely small. (For the sake of argument, we will ignore relativistic effects, which compel us to take more care when contemplating concepts regarding simultaneity.)
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In this paragraph two sidetracks are discussed to anticipate on questions/thoughts/considerations which might arise pertaining extraterrestrial civilizations and the origin of our Sun.
Sidetrack 1: Speculations about very old advanced interstellar civilizations
One might speculate about the existence of extraterrestrial civilizations which have found ways to break free from their home planet and have made themselves independent from its fate. Civilizations which have the technology of engineering themselves to adapt to various environments, to preserve genetic diversity and to spread throughout space. Civilizations which have found sustainable paths towards hopeful futures and have found ways to successfully avoid paths which will lead to destruction. It can be expected that advanced civilizations might have become somewhat nomadic in nature on an interstellar scale and might have a very long lifespan.
However, one might wonder what such civilization might look like. A civilization, over one billion years old, which might have become alienated from its home planet a long time ago, whose home 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 endeavours? Maybe such civilizations might be very alien to us. Our current state of humankind is far from suitable to build a very long-term sustainable future here on Earth. We might be even totally insensitive to their state of mind on a very fundamental level because it might be inherent to our human nature, not able to comprehend the required delicate fabric and the delicate state of such advanced civilizations.
We also need to realize that having a perspective on an interstellar future is not something all living creatures derive purpose from. For example, for the vast majority of us it is enough to find purpose in this life here on this beautiful Earth of which we are an integral part. Just an awareness about the existence of a large universe beyond Earth is the icing on the cake for this vast majority. So civilizations that went interstellar might therefore also have resulted in new planetary civilizations from which traces to their origin have been wiped out deliberately at their initiation. The result is civilizations that have been detached from their interstellar origin deliberately. Perhaps because the purpose of this interstellar connection was deemed unnecessary when sovereignty had been given, respected, cherished, and protected by their parent civilization.
On the other hand, our history shows that developments in science and technology are being accelerated during wartime. The marvellous 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 interstellar-faring civilization had to go through. Such a civilization might have been forged by struggle and rivalry and, to a much lesser extent, by spiritual development. 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 when engaging in contacting such a civilization.
However, speculations about very old interstellar civilizations still belong to the realm of pure fantasy. It is very quiet out there. Which leads to the conclusion that if the origin of life and civilizations is probable in this vast universe or even in our own Milky Way with its estimated number of 100 billion stars, the chance of two civilizations existing simultaneously is extremely 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. The emergence of civilizations in our Galaxy are brief isolated events randomly spread throughout a vast sea of time.
Sidetrack 2: Brief remarks about the origin of our Sun
When we look to the composition of our solar system, which includes our Sun, we find heavy elements of which we have learned how they are formed during the life cycle phases of heavy stars. Various observations have revealed how all the chemical elements are formed. Many chemical elements are formed in or near stars out of lighter elements that have been formed in previous generations of 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 very heavy stars, like what is happening in the Eagle Nebula 7000 light-years from here. It is therefore also likely that our Sun was once part of a group of stars which emerged from this nebula of remnants of one or multiple supernovae. It is assumed that this nebula, this cradle of our Sun, has been dispersed over the past 5 billion years throughout our Galaxy. The search for remnants 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.
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