This debate is really quite a toss up. There haven't been any conclusive evidence to say whether Neanderthals were able to speak the way Homo sapiens could but since there were studies that showed they interbred, maybe it was possible that they communicated with each other albeit not in the way we expect.
Discover Magazine's Bridget Alex writes:
Part of the reason scientists disagree about Neanderthal language is because there are different definitions of language itself. Without straying too far into academic debates over the nature of language, let’s just say there are broad and narrow theories when it comes to what actually constitutes language.
Speech and language are mostly soft-tissue operations, requiring organs like the tongue, diaphragm and brain that rarely preserve. However, producing and hearing speech influences some enduring aspects of our skeletons too, including the hyoid bone, ear ossicles and the portion of the spinal canal that holds nerves involved in precisely controlling breathing. Studies have found these features are very similar between Neanderthals and Homo sapiens, but more primitive and ape-like in earlier hominins like Australopiths.
The question of Neanderthal language remains an open debate. If they lacked it, language may be unique to Homo sapiens. If they had it, language was likely present at least since Neanderthals and modern humans shared a common ancestor, over 500,000 years ago.
So which team are you on?
(Image credit: Wikimedia Commons)