We can probably learn a lot of the power usage from the 295X2, which was two 290X chips "slapped" together. Each 290X uses about 295W peak, but the 295X2 uses about 500W peak. AMD gave a pretty detailed requirements list for the 295X2, stating it needs 50A on +12V, and they suggested a few PSUs for it -- all 1500W or higher, but it's very clear that the card uses nowhere near 1500W.
Here they've recommended 1000W PSUs, and a 390 8GB card uses 250W peak, so naively we'd expect this to use 500W peak. However, with the 295X2 we would've naively expected 590W and it comes in at 500W, so we may well expect a power savings here of 15% or so, bringing us to around 420W or 35A +12V.