Well, some of you are on the right track. Power out = Power in minus the efficiency loss of the driver circuit. A 2 cell light should last twice as long as a one cell light..........if the output current to the LED's are the same. It most likely is not. That's the problem both the LD12 and LD22 are stated at 3 lumens but one of them has to be wrong because the math wouldn't work out. I usually don't get too worried about these things because I have tested many lights with a data logging light meter and pretty much everything comes into balance from an engineering perspective.
Let's take lightwait's measurements because that looks to be reality since he measured it. I'll estimate the batteries to be 1.35V, the forward voltage drop across the LED at around 3.15V, and the efficiency of the driver circuit at 85%. I don't think because the LD22 has two batteries vs the LD12 with one that one will have much of an advantage over the other. It really does boil down to power out = power in minus the efficiency of the driver circuit. The LD22 simply has twice the power at the input. We don't care what the voltages are because boost circuits are typically 85 to 90% efficient and the driver cirucuit is going to bump the input voltage to whatever is necessary to drive the LED at the desired regulated output.
Here's an example of how all the calculations stack up. Run time simply would be the 2000mA capacity of the battery divided by the current being drawn from them.
LD12 run time: 2000mAh divided by 29mA = 69 hours
LD22 run time: 2000mAh divided by 19mA = 105 hours (keep in mind that 19mA is coming out of both batteries since they are in series)
LD12 Power in: = .029A x 1.35V = .039W
LD22 Power in: = .019A x 2.70V = .051W (from this alone I can tell that the LD22 would be brighter because its taking in more power from the batteries. And I can tell the LD22 should last longer because it has 2 batteries but the power being consumed by them is not twice the LD12's)
LD12 Power to LED: = .039W x 0.85 = .033W
LD22 Power to LED: = .051W x 0.85 = .043W
LD12 LED current: .033W divided by 3.15V = .0100A
LD22 LED current: .043W divided by 3.15V = .0137A (based on lightwait's measurements, the LD22 should be about 37% brighter assuming linear relationship of lumens to current)
These are theoretical calculations. To be absolute, actual measurements would have to be made. Plus the battery voltage doesn't stay constant either. So this is a rate of change calculation with differential math but you can pretty much estimate things the way I did above.