OK, so I just connected a 1 Ohm resistor in series with one of the modules, then measured the voltage across the resistor as 0.08V.
Ohm's law says that
I = U/R
So that makes for 80mA of current through the module. The voltage across was only about 10V, which makes for about 0.8W for that module. This was for a cool white with ~10000K color temperature because that's what I had laying around
The warm white have on the order of 3000K color temperature, and I anticipate them drawing a little, but not significantly less current. Looks like the specs aren't too far off - tolerances notwithstanding, it seems 0.7W is pretty close for the warm white LEDs.