Do they have functioning jaws? Yes. Although warforged don't need to eat or drink just to survive, the rules state that they are capable of eating and drinking (for example, to use a potion or benefit from a heroes' feast). It's not hard to imagine a warforged drinking without jaws, but eating without jaws is much more of a stretch.
Do they speak by moving their jaws? No. Warforged may have jaws, but they don't breathe, so they have some other means of producing sound: a speaker-like system, as you suppose. But that doesn't mean they can't move their jaws when they speak; it only makes it unnecessary. It could still be part of the core programming of a warforged, as a cosmetic UI-type thing. Even if it isn't, they could still choose to do it.
Do they move their jaws when they speak? The rules don't say. Maybe the novels do. Having not read them, my personal interpretation is that it depends on the attitude of the individual warforged. If they want to put humanoids at ease (or are actively exploring their living aspects), then they move their jaws just like living humanoids do. If they've embraced the machine, or just don't care what humanoids think, then they don't bother to move their jaws.