Most of our religions already have been through contact with another intelligent civilization, in 1492. They adjusted to that in various ways, and would probably do something similar as a reaction to contact with an extraterrestrial civilization.
Years ago, I think in relation to some probe launched to Mars, a short article in my paper mentionned that a commission had met at the Vatican to discuss the hypothesis of the discovery of intelligent life somewhere in the universe.
The general position of this comission was that if ET races exist, they must have had both an original sin and been visited by Jesus, in one form or another.
This amazed me, not only the fact that they had actually considered the issue, but also because I had just read Clarke’s “Meeting with Rama” (??? I’m not sure of the original english title) where a catholic astronaut sent to investigate a potential ET ship get to meet the pope to have his opinion about the religious consequences of the existence of another sentient specie. And the pope, in the novel, gives essentially the sama answer as above. If there are ETs, they had their own version of the original sin and the savior hence have the same “religious status” as humans. I was amazed that a SF author guessed right on this topic.
Of course, the conclusions of some random commission doesn’t prove that this will actually be the official stance of the church (let alone that catholics will swallow that), but it shows at least that the Vatican plans ahead.
So what?
If FTL travel remained impossible, political/diplomatic relations with the ETs would be meaningless. No one on Earth would be in a position to make alliances with them, fight wars, establish trade agreements, or exchange anything but information. And the world scientific community could handle that. No one else would be qualified.
So, there would be no point in sending them missionaries?
You hoped to apply for a space trip paid for by the Vatican? 
I’m sure most everybody would be qualified to give his opinion about what kind of infos can/cannot be exchanged. So, politicians would have to be involved.
Hah! I just saw this movie the other day. Not as good as the book, of course, but okay. Jodie Foster did a passable Arroway.
Anyway, back to the OP.
As far as verification, SETI, or whichever agency discovers the signal, would presumably, once confirming a predetermined threshold of non naturalness has been met, attempt independent reception (from another facility).
If the signal contained some kind of message I don’t think we’d easily detect it. But let’s say we did. It would still probably take years, if not much longer, to make any sense of it whatsoever…though that would be the fun part.
Since it’s hard for us, as humans, to think “outside our own heads” I believe we’d have a very difficult time decoding even the simplest of messages. We’re wired one way, and tend to make presumptions that other living entities must be wired the same way we are when it comes to expressing concepts like mathematics, physics, etc…, as though there’s a universal norm for such things; 0, followed by 01, followed by 10, followed by 11 may only make sense to us.
As far as religion is concerned, you’re darned tootin’ there’d be an effect. There’d be such a rapid metamorphosis of existing dogma and the development of new dogma incorporating the new reality that the very tenets and definition of religion would probably be altered, almost overnight. It’d actually be very interesting to see how quickly Christianity co-opts the existence of non Earth-based sentient life and begins to reinterpret its gospels, not only as proof that alien life was written about in the Bible, but also as celebration of the purported undeniable evidence of God’s existence through another of his wondrous works revealed.
I’m sorry to say that I believe the effect on society would be anarchic. Let’s face it, even if the message came from multiple hundreds of light years away, such scales mean nothing to most people, and the general reaction would probably be fear. I think the world at large would experience a period of subdued panic, which would effect the global markets as well as individual productivity, with people second-guessing their lives and drastically shifting their priorities. I don’t think it’d be pretty for a while.
I know that they believe that the former is true of the Native Americans, Aboriginal Australians, and other peoples who had not heard of Christianity before contact with Europeans, but do they believe those people were visited by Jesus in one form or another prior to contact with Europeans? That’s news to me, if so.
I do not find it surprsinig at all that the church is having these sorts of discussions internally. To this day the church pays for its handling of Galileo. Prior to Galileo science and religion got along and even supported each other. After Galileo it seemed the church’s dogmatic viewpoints were really not something science could coexist comfortably with.
I think the church (I use the term “church” here meaning the Roman Catholic church but I believe many other religions are in the same boat) has learned where it can delineate the lines between science and religion and religion will stick to the “faith” areas where science cannot really intrude.
In Stephen Hawking’s book A Brief History of Time he relates a story of having an audience with the Pope. The church was apparently interested in his work on the beginnings of the Universe as it approached the moment of “creation”. IIRC Hawking, who shared a birthday (or something like that) with Galileo, was a bit worried when the Pope told him that his work was fine except that he should not look into the moment of creation itself as this was the province of God. Hawking decided against telling the Pope of his most recent speech where he described a Universe with no place for God (or something like that).
I think the church now pays fairly close attention to the happenings of science and discusses them at length. Finding an ET species would certainly generate many questions for the church. Better to have had the discussions out over several years rather than jump to hasty decisions.
Most of the posts in this thread assume that we will have detected intelligent signals and can read their context well enough to gain detailed information about the senders. Rereading the OP however, I decided to back up a few steps. Here’s some other scenerios:
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Ambiguous Signal. A signal is detected, and it looks like it might be artificial, but it’s not immediately clear what it contains and a few holdouts maintain that it could be caused by a natural phenomenon.
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The Teaser. A definitely extraterrestrial artificial signal is detected and data is gathered for a short time- a week, maybe a month. Then the signal disappears and is never heard again. All we know is that something once created a signal at X-Y coordinates, and the limited data is reanalyzed and debated forever.
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Artificial but No Data. We detect the equivalent of an unmodulated radar beam.
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Artificial but Undecodable. Signal clearly contains data but we can’t figure out the context.
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Artificial, partially Decodable, non-Deliberate. We overhear a signal that wasn’t deliberately sent to us- “leakage” from their radio, tv, etc. We can figure out at least the format of the signal- whether it’s sound, pictures, data, etc.
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Deliberate Hello Message. Contains clues as to how to decode it- number sequences, etc.
Scenerios 5 and 6 are the only ones where we learn much more than “someone’s out there”, which a lot of people assume anyway. A lot then depends on just what the aliens were like. We might be very surprised at just how different from us they could be. One extreme example is non-intelligent life that evolved, not developed, radio.
I was personally taught that as a child. It was one of the things that soured me on religion. I was told it was because we were made in God’s image; therefore any non human being must be Satanic.
I think it would be fun to watch the denials from the YEC if we contact a race with records going back, say, 10,000 years.
!!! What kind of anatomy would that require?
The Mormons believe that after Jesus rose from the dead he visited the Nephites (an offshoot of the Israelites) in the Americas. However, the Nephites were later destroyed by the Lamanites, so they would not be considered the cultural or biological ancestors of any of the Indian nations.
It’s not all that unlikely. Animals on Earth already use electricity, and radio is fairly low powered. It’s not like postulating organic nuclear fission.
Then why has it never happened here? The ability for animals of a given species to communicate with each other by radio waves – completely imperceptible to other species – would confer a significant survival advantage. But AFAIK, electric eels, for instance, cannot use radio signals to find mates or coordinate their hunting or whatever.
Haven’t we already had a few “practise runs” with the belief that beings already exist on other worlds, (for example the “canals” on Mars?)
Wouldn’t the reaction to actually coming into contact with an extraterrestrial civilisation today, be similar to the acceptance there seemed to already in place, when people believed there was actually life on Mars?
Chance. Evolution has no foresight; no plan. Unless the right mutation happens, and the original mutant survives and breeds, evolution will never explore the possibilities opened by that mutant.
All your word are belong to us. 
Well, it was only a theory then. And pretty much discredited in the early 20th Century. Since then, the cultural importance of science fiction has grown enormously, and it has dealt with practically every possibility concerning ETs – that they would be indifferent to us, that they would want to conquer or exterminate us, that they would want to help us advance, many others. There’s a lot more, now, for the public imagination to work with.
{Answer assumes a signal, no inclusion of how to build a FTL or near LS, no information to build new Technologies, no Hitlerian threats, no way they or us will meet face to face for decades, if ever}
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We would never have another serious SDGD on whether NASA’s Budget/Space Exploration was worth it. NASA’s Budget and any associated technologies would receive increasing funding levels for the foreseeable future.
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Science funding of all types would see post-Sputnik like increases
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Other Countries would begin crash SETI programs to either talk to or listen in on the conversations or seek others. Countries like North Korea, Iran, Syria, or Cuba might opt out of a UN frame-work and rapidly build huge Arecebo-style radio telescopes (if that was how it was sensed - or Very Large Arrays or …). Countries like China, India and Russia might begin to put crash plans in place to put dishes on the moon or orbit (or whatever).
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Some sub-set of Christian Fundamentalists would accept that the creation story isn’t meant to be read as literally true, and that a theology with Humans not as the Center of God’s Universe would be developed. A theological change every bit as unsettling as Copernicus was to the view of the Solar System. But Fundementalist Christianity would survive more or less as we know it even among the sub-set that no longer claims every jot and dot in the Bible is literally true.