While it may be true that nonhuman primates do have the ability for rather sophisticated communication, their language structure has never be demonstrated to even come close to the complexity of human language. Human language universally involves complex rules which even those celebrated apes (Nim Chimpsky, Washoe, Koko, Sarah etc.) trained to sign were never able to reproduce.
Guess that means we don't have to worry about them animals talking behind our backs about us.
There is obviously a gap between humans and other non-human primates. However, Savage-Rumbaugh's research with Kanzi (a language-trained bonobo) indicates, at least, that Kanzi can comprehend some grammatical utterances and relationships. She also suggests (although not very conclusively, in my opinion) that Kanzi also produces some grammatical utterances. The gap may not be as wide as we think.
Savage-Rumbaugh, E. Sue, et al, eds. Language Comprehension in Ape and Child. U of Chicago, 1993.
Greeenfield, P. M., and E. S. Savage-Rumbaugh. "Grammatical Combination in Pan paniscus" In "Language" and Intelligence in Monkeys and Apes: Comparative and Developmental Perspectives. Edited by Sue Taylor Parker and Kathleen Rita Gibson. New York: Cambridge U Press, 1990.
Department of Religion