Just out in Frontiers in Human Neuroscience is a short opinion piece on whether speech is processed mostly in both hemispheres, or preferentially in the left hemisphere. The bottom line is that it depends on whether "speech" refers to speech-like sounds, words, or connected speech (e.g., sentences). Although this might seem an obvious point, it's one that fairly regularly gets glossed over in the literature.
It's a short read, but there are two main points worth emphasizing, one more theoretical:
Phonological and lexical information is processed largely bilaterally in temporal cortex, whereas connected speech relies on a left-hemisphere pathway that includes left inferior frontal gyrus. Importantly, the distinction between unconnected and connected speech is not dichotomous, but follows a gradient of laterality depending on the cognitive processes required: lateralization emerges largely as a result of increased linguistic processing.
and one more methodological:
[F]or true claims of differential hemispheric contributions to speech processing, the left and right hemisphere responses need to be directly compared...In the absence of these or similar statistical comparisons, any statements about lateralization of processing need to be made (and taken) lightly.
See the Frontiers website for the whole article.