Ok, I am going to give this a go...
By training, I am a philosopher and theologian, not a scientist, so I may look at things a bit differently than most. More to the point, the burden of proof I demand for anything I affirm is exponentially higher than the scientist would. A scientist is typically satisfied with a theory that is internally consistent and matches the available physical data. People of my background typically demand that it not only does this, but matches the data (not just physical data) in ways better than any hypothetical alternate explanation. This is why philosophy goes slower than science (it may take a couple hundred years to formulate the alternate explanations), but can address questions which science is incapable of addressing. Not that there is anything wrong with science, but every field has its limits.
All that to say, things like the Drake paradox are unvetted, and quite honestly poor, philosophy under the guise of science, at least when it is used to establish that there is no intelligent extraterrestrial life. There are myriad possible reasons we haven't discovered them yet, so this confuses our ability to detect something with its actual existence. Any philosopher worth his/her salt would cry foul, unless they held to the belief that reality itself is relative, but this is not a widely held position (for good reason). All that is required to defeat the paradox in logic is to posit that there is a plausible reason we have not detected them (what that reason is doesn't need to be defined). The entire paradox falls apart with that, so I reject that paradox as drivel from the outset.
But, this does not mean that intelligent extraterrestrial life does exist, either. Simply dispelling an argument against it does not constitute an argument for it. We must suspend judgement until there is evidence one way or another, and being this is dealing with the physical universe, the evidence which should be sought is scientific. But, given the barriers present to detecting this, I wouldn't be surprised if humanity never discovers life outside of this planet. As such, it remains a source of opportunity for SF writers, either in denying or affirming their existence.
Now, religious texts have been brought up, presumably the Bible (though it could include the Koran as well). I only wish to touch briefly on this here, and will happily discuss this further in PMs if anyone is interested, but there is nothing in the Bible which addresses the question of extraterrestrial life one way or another, either explicitly or implicitly. It does mention humanity being in some way Imago Dei, but there are three major theories on what is intended, and only one of them has anything to do with cognition (and it is not broad intelligence, but moral agency which is in view). But, this does not preclude life, even intelligent life, elsewhere. So, while some armchair theologians may argue otherwise, the discovery of extraterrestrial life is in no way incompatible with the Bible. Any other discussion on this topic I would prefer to keep off the main forum, as this easily leads to arguments.
So, I think we should adopt an agnostic attitude toward alien life, and create worlds where one or the other option is true.