Update, Dec 2016: Just this month, Michigan legalizes self-driving car testing on public roads, and GM is going to start testing. As things continue to move in this direction, the field will no doubt start to lend itself towards developed communications protocols.
At the same time, also this month, the NHTSA is pushing to move forward with deployment of mandatory V2V protocols, which ultimately will make the answer to "Do self driving cars talk to each other?" a solid yes.
Like I said below, we're right on the cusp of this technology. It will happen.
From Sep 2016:
The short answer is: Not yet. But eventually.
Right now, I know the Teslas don't. AFAIK the Google ones do not either.
First of all, some concrete info: At least as far as the NHTSA is concerned, a few years ago they started murmuring about a potential vehicle-to-vehicle communications standard, at least in the US (V2V). The project web site is at http://www.safercar.gov/v2v/index.html and you can check there for updated info. At the time of this writing, the latest development is that meetings are still being held and the project is still in the assessment and planning stage. A recording of that meeting from April 2016 is available. This means that at least in the US, there currently is no standard communications protocol for it.
On the more conceptual side: Ideally, you would not want self-driving cars to have to talk to each other in order to maintain safety in today's conditions, because most cars are not self-driving. Ideally a self-driving car should avoid collisions with other vehicles and objects on the road, regardless of whether or not those other vehicles are also self-driving -- it shouldn't have to know or care what's controlling other vehicles. That is, other vehicles are simply other vehicles to be avoided, rather than special cases.
This means that, at least today, if a self-driving car was unable to reasonably avoid collisions with another vehicle unless it was also explicitly communicating with that other vehicle, then really that car would not be suitable for safe road use.
You also don't want to rely on it because if e.g. there's some atmospheric interference or damage to the communications, you don't want cars to no longer be able to avoid collisions.
Also, in the future, at least IMHO, car-to-car communication would be extremely desirable not just for collision avoidance but for traffic management as well. For example, here is a video discussing autonomous traffic management protocols, as well as the home page of that project which contains many interesting references. The page is a bit old and some links are broken, I remember they used to have a simulator, circa 2004, but I can no longer find it, not even on archive.org. But, I digress...
There's still a long way to go to get the vehicles to be as safe as possible on their own, and there's really not a lot of them. Autopilot now still relies on a lot of care and monitoring on the part of the driver to maintain safety; for example, here is a video of a complete 45 minute commute with a Tesla system, driven by an attentive and responsible driver who understands its limitations (example), with commentary along the way. It's a long video, but a fascinating look at the current strengths and weaknesses and a good look at what autopilot is currently like for the vast majority of us who don't own these cars ourselves.
All of this technology has been in people's minds for a very long time; we've got the building blocks now (well-developed machine vision algorithms and such) and it's only relatively recently that the self-driving and control algorithms themselves have gotten to the point where we can start using these cars on the road. The basics are in place, but still need polishing, one step at a time. The current focus is just getting these vehicles functional, but discussions are certainly underway about vehicle communications in the future, because the need for that is now just around the corner.