Suppose a person reads a book out loud for 1 hour straight. Would this "exercise" the parts of the brain involved in speech? Would this make the person a better communicator in social situations?
Based on general principles of skill transfer:
Based on the same principles, if you want to be a better communicator in social situations, your time would probably be better spent (a) studying social interactions, and (b) engaging and reflecting on social interactions.
Update: @ofri makes a good point. If a bottle neck on your social skills is actually articulating words clearly, then practicing that skill, such as would occur when reading, should be effective. Ofri uses a good example of a second language learner. The case I'm thinking of above is where the person is fully competent in articulating words verbally, and where problems with social skills relate more to knowing what to say when.