How did the Paleolithic humans talk to each other?

The question is somewhat difficult to answer. The term "paleolithic" does not actually refer to a specific period but rather a stage of technological development achieved by different cultures at different times. The Jarawa Tribe of the Andaman Islands used paleolithic technology into the twentieth century whereas Mesopotamia underwent a neolithic transition ca. 9000 BC.
In terms of languages, it is really impossible to know how people talked before the invention of writing, a technology that preserves speech. The best we can do is extrapolate earlier languages from later ones using what we know about the dynamics of language change. For example, in some languages (Greek is a major example) there is a reduction in the number of verb moods and tenses and noun inflections over time, but that is not a universal phenomenon.
In general, high mobility and larger communities correlate with faster language change while smaller, more isolated communities tend to preserve their languages intact, and thus we can assume that the rate of language change was slower in paleolithic than neolithic communities. One example of a surviving paleolithic language which has been recorded by modern technology is the click language of the San Bushmen. This type of "click language" may have been more common among our early African ancestors than in more modern communities.