H,, this first one (PowerGen) looks suspicious because there's only a note that it can charge two items at a time, but nowhere stated, that it can "Charge two AND be charged in the same time"
I own one 5600 mA but I realized it can work that or that after I opened the product and read manual.
That's why I would like to be sure I won't buy another unneeded (=poor) charger.
You're saying you own that first one (PowerGen).....?
(Sorry for the slow reply, too busy playing with 2Pi's & an Atrix Lapdock, & only slowly getting round to draining & recharging the battery pack...)
... Yes, I have the Powergen one, & YES! it does charge itself whilst outputting thru either or both of the USB outlets. I've been using it as a "UPS for Pennies" on the Pi, but have found it so useful as a carryout (particularly with the 360-hr+ LED "torch") that I keep disconnecting it, & the Pi, to take with me on my travels, rather negating the benefits of having battery backup for always-on Pi as the whole lot needs unplugging to put the Pi back onto direct power!
One small caveat (this caught me out for a while as I was using a standard eBay/Samsung mobile charger to run the Pi originally) : This scenario *WILL NOT WORK* *UNLESS* you use the 5'3v supply which comes with the battery pack: makes sense really; if you're expecting the pack to supply a (two) healthy 5v outs @ 2a, .6A, there's no point trying to simultaneously charge it from a standard 5V usb supply or port: you *need* to feed it with a slightly higher voltage to offset the internal losses & actually dump some energy into the batteries. Please also note that whilst the pack is perfectly happy to supply at least 2.6A combined output, the micro-USB input and charger supplied are only rated at 1A maximum, so if you are drawing any more than about .9A in total, the battery *WILL* discharge slowly, even though it is notionally charging.
All that said, I've been so pleased with the performance using the original pack, & so miffed that I keep diverting it for my own perverse purposes, that I've now gone & bought a an Anker "Astro" 10,000mAH pack (Amazon again, abt £40) which surpasses even the PowerGen one (Although it's considerably bigger & heavier, won't charge from standard/micro USB (comes with its own 12v jack plug charger), & lacks the LED torch, it has an additional switchable 9/12V jack plug outlet which means that for me, it covers just about every emergency power rquirement I can think of - I've even been using it to power 19v laptop car adapters & a 240V inverter, just for the hell of it !!). Haven't been able to run a conclusive "will it fully recharge from nearly empty whilst supplying power out" test yet, but the charge LED's flash in this scenario, & it seems to have happily gone from 1/2 to 3/4 power in a few hours whilst feeding my Pi.
(Incidentally, the 2600mAH solar pack I alluded to in my first post also charges-whilst-supplying, either from it's mini-USB power in port, or from all this glorious Sunshine we've been been getting ... NOT!)
Hope that helps !