This is a great question, but unfortunately no one will be able to give a definitive answer as to the chances.
This is simply because we do not yet know if life exists anywhere other than Earth. It is still within the realm of possibility that Earth is the only place where any kind of life, what to speak sentient life, exists.
Finding such life would be an affront to many religions that contend that God created life on Earth only, and therefore finding even a fossil of a microbe on another celestial body would be the holy grail of science and have reverberating effects throughout human society. The theory of evolution would be proven.
It is believed to be highly likely that life does exist elsewhere in the universe simply by the laws of probability, but that being said, we have found no evidence to support this theory.
So now we can come back to your original question: “What are the chances of two planets from the same solar system having sentient species that reach space exploration around the same time?” Based on the fact that life has not been discovered anywhere, the chances to sentient life forms existing in the same solar system is practically nil (but not impossible).
Are octopuses originally from another planet? It sure seems so, doesn’t it? They are a very weird creature!
But just because they are weird doesn’t mean they’re from another planet! In reality they are related to several other Earth critters. They are a mullusc and belong to the class cephalopoda along with squids cuttlefish and nautiloids. So although they are very weird, they are definitely not from another planet.
However, in the sense that we are all made of stardust that came from some Supernova explosion somewhere in the universe, then we are all from another planet or star!
Do you think aliens have received our communications and chose not to respond? Well, who knows, but…. radio waves travel at the speed of light. We’ve been emitting radio waves for maybe a hundred years. A hundred light years radius around our solar system doesn’t encompass a lot of stars. So the answer to your question is “probably not.”
Pretty much all of science and scientists in the world believe in extraterrestrial life, even though no evidence of extraterrestrial life has ever been found to date. Why do scientists believe in extraterrestrial life? Simply because there are billions of stars in the Milky Way galaxy and billions of galaxies in the universe. Somewhere it is highly likely that extraterrestrial life exists. It is even possible that extraterrestrial life of some form or another exists on Titan, within our own solar system!
An alien invasion by secular and militaristic extraterrestrials would solve one of the biggest problems we have here on Earth. That problem being: does life exist on other planets? Finally we would have an answer!
I don’t see either as being dependent on the other. In other words, you can believe in God and not in extraterrestrial life. Many religious people do in fact hold that view. On the other hand, you can believe in extraterrestrial life and not believe in God. I know many people who hold that view.
So I don’t see a paradox at all. According to the Google dictionary, a paradox is “a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true.” I fail to see a contradiction that two different people can hold either view. It’s all theory.
In reality, we do not know for sure that life exists anywhere other than Earth because we have never found any evidence of life, although we continue the search. We have yet to find even a fossil of a microbe from space. However, the laws of mathematical probability suggest that with billions of planets in this galaxy and billions of other galaxies, it is likely that there is at least microbial life somewhere, if not something more advanced. We just haven’t the proof yet.
Similarly, except for subjective experiences, there is no proof that God exists either, although people believe that God exists. Maybe both exist. Maybe neither exists. I suppose it’s a question of faith in either view.