Google and Facebook give soldiers intelligence products? Or Palantir operates websites and does analytics of their users? I don't think I understand the comparison.
I worked for a competitor to Palantir at one point. Basically, they're using what you can think of as an adaptation of Google's PageRank- instead of ranking how important websites are by evaluating the links between them, Palantir and the like estimate a person's importance / influence by evaluating their connections to other people.
You can do some pretty interesting market analyses for commercial purposes using just data from Twitter, or Facebook if you can get access to it. What Palantir can do with covertly / illicitly gained data is much the same, only with much deeper wells of information to draw from.
That depends entirely on the product and market, though there are a few general uses. Generally speaking, find the people who are very passionate and vocal about topics your product addresses.
Now, you know two things: the types of words they use, and with a bit of digging, the sentiment typically associated with those words. This can influence your marketing copy and promotions.
Repeat with a few demographics to identify associated areas of passion. Now, you have a group of very passionate people, and maybe even a loose network. Dig in and see if you can find more common associations to tighten the network up a bit.
From there, you could run a targeted campaign- say, a contest with freebies awarded to a number of people who retweet a certain message or hash tag.
People who are tangentially interested hear about your product. People who would have heard about it anyway (by virtue of being in your target market) will now be talking about your product, using your specifically chosen wording.
For the cost of a few freebies, you now have reached your target audience- and thousands more, if they have a significant following. Further, you're not spamming airwaves, or cold emailing, or running ads that are blocked or ignored anyway.
Take all of this with a hefty dose of salt. I'm an engineer, not a sales person or marketer. I'm not on Twitter or Facebook or any other network, so I don't really know all the ins and outs of how to use the product itself, only how to build things those people found useful.
EDIT: for what it's worth, you can do all of this by hand. Doing it well, doing it right, is harder. I won't recommend anyone specifically, but if you're interested in marketing this way, I'd suggest finding someone who already knows how to calculate influence very well and using their services. Otherwise, you'll spend a lot of time on a pointless goose chase, because there is a LOT of noise to sift through to find a good signal, so to speak.
You can do some pretty interesting market analyses for commercial purposes using just data from Twitter, or Facebook if you can get access to it. What Palantir can do with covertly / illicitly gained data is much the same, only with much deeper wells of information to draw from.