A lot of work is being done to completely sequence the Neanderthal genome. One of the groups involved in that genome project has just published a paper in Current Biology reporting that Neanderthals had a modern version of the FOXP2 gene, a gene that has been linked to speech and language ability. There are two point mutations in this gene that are the difference between humans and chimpanzees. The Neanderthals also have these same two point mutations.
This same research group has reported they find no genetic evidence that Neanderthals and modern humans interbred, and they report this again in this new paper.
But it's fun to speculate whether Neanderthals and modern humans were able to communicate with each other. Obviously, among Neanderthals, the entreaty, "Don't eat me!" had no effect, because Neanderthals, with all of their sophisticated culture, were cannibals.
Here is the NY Times take on this research, another from National Geographic online, and the Los Angeles Times article, which mentions the cannibalism.