We do but it's unfortunately not widely covered.

The Snowden document released on the strategic objectives - the high level goals driving all NSA technology - included strategic communication, which we know from DoD investment in the last two presidents has increasingly been focused toward social media (with some programs like SMISC and MINERVA as public research instanciations of the policy).

We know quite a bit about the US attempted partnerships with Twitter (historically Twitter has refused partnership but now is more congenial) from FOIA documents and news sources and that the US uses sockpuppets in many places, including Twitter for its own propaganda purposes worldwide. We know that the Facebook studies on nudging voting behavior and emotions had ties to these programs as well.

We know that threat intelligence systems include patterns to block certain URLs and content that is foreign controlled propaganda - notably of Russian origin and that there are people in the government full time on tracking Salafist ideas online for both blocking when they reach American audiences and engagement of various sorts overseas.

We know that there is a firewall between the State Department and DoD when it comes to the programs (State Department has the overt programs for the most part and the DoD the covert). We know a few things from various leaks about partnerships (HB Gary Federal, etc) with private sector intelligence firms.

We also have reports from the US and journalists about some of the techniques of adversary nations and we have some details especially about GCHQs JTRIG capabilities and some of the areas the NSA and GCHQ have worked in due to Snowden document. We have a couple of other examples - things like ZunZuneo - that some social media platforms are fronts for various countries rather than merely partners.

We also know some things about the law in the United States for messaging: foreign messaging the US has pretty much no limitations. While it will censor American political speech (al-Alwaki's youtube videos, etc) for the most part it can justify minimal amounts of this and focuses on foreign speech and foreign aimed influence.

It is legal in the United States to message Americans under wartime powers, states of emergency (Occupy and Ferguson were both considered states of emergency), and for the purposes of getting domestic support for overseas military operations. The Smith-Mundt Act was recently revised to remove the limitation on the US government to prevent foreign-aimed propaganda from being incidentally consumed by Americans (the internet makes this hard) but it's unfortunate that this happens quite a lot (why they had to make the revision) and that it has, in the past, been purposefully abused.

We also know some things about historical partnerships, but most of that has to do with 'old media' and it's hard to draw real policy and technical implications from programs that existed a generation ago. The things we know about those partnerships today (partnerships with media executives, leaked editing of journalism coverage) that look like the old stuff don't focus on social media so much - the social media programs are a relatively new initiative.