Adam Frank and Woodruff Sullivan have published a paper  riffing on the famous Drake Equation, which sketches the factors that could give us an estimate of the number of “technological civilizations” that might be out there today. His heuristic includes factors for the number of stars, how likely “intelligent life” is, and, most tellingly, how long “civilizations” last.
To date, we can’t really do more than guess about most of his parameters.
Frank and Sullivan go a different way with this idea. Forget about “today”, what are the probabilities that “civilizations” could have existed ever, during the life of the universe? They argue that this question is simpler, especially by eliminating the extremely difficult to estimate “lifetime of a civilization” factor.
Basically, they group things together into two factors, how many habitable planets have there been, and how likely is “intelligent life”, given that you have a good planet. The second term is completely unknown, but we do have a decent estimate for the first—and it is “a lot”.
A = N x f
Their point is that for any given number of civilizations, we can calculate the threshold probability that would balance the equation. The first question is, are we the first an so far only civilization? Their calculation says that only if the probability of intelligent life developing on a habitable planet is less than 2.5 · 10-24 would we be likely to be the only intelligence, ever. e.g., for probability 0.01, plug in an estimate for the number of habitable planets:
f = N / A
f = 0.01 / 4 x 10^21 = a really small number!
And so on.
The upshot is that, assuming that we understand things more or less correctly, it is likely that many other civilizations have existed. How many are out there now? That is a different question, and I recommend Webb’s book, “Where Is Everybody?” 
My first reaction to this paper was, “Of course!”. It just seems to fit my intuitions.
I realized that the reason it seems so natural is that it is really just banking on the fact that there are a lot of stars out there, and a lot of time, and in the last decade we have come to know that there are a lot of habitable planets, too. Given enough time and space then, unless life and intelligent life are a total fluke—and 10^-24 is pretty darn flukey—it must happen, and must have happened.
Of course, by the same logic, even if there are many interesting things, including “civilizations”, out there, there is a whole lot of “there” there. It could take a lot of time and work to find the interesting needles in the giant haystack, even if they aren’t deliberately hidden. And always assuming we even know how to look.
- Adam Frank and Woodruff T. Sullivan, A New Empirical Constraint on the Prevalence of Technological Species in the Universe. Astrobiology, 16 (5):359-362, 2016/05/01 2016. http://dx.doi.org/10.1089/ast.2015.1418
- Stephen Webb, If the Universe Is Teeming with Aliens … WHERE IS EVERYBODY? Seventy-Five Solutions to the Fermi Paradox and the Problem of Extraterrestrial Life, New York, Springer, 2015. http://www.springer.com/us/book/9783319132358