Yep. There's no confusion over it or "different answers". The specification says 70mA per chip.
Per pin, in what I linked.
The pins you're referring to are "Vcc" and "GND", not I/O pins.
It seems an odd qualification to make, since there is only one Vcc pin and only one Gnd pin.
I'm puzzled now about how my LED sign is working. I presume it was designed by professionals. At a given moment one 595 can be driving (sourcing) 8 LEDs, via a 150 ? resistor (each), and running at 5V. From my calculations above that should be 20 mA of current each, so that would be
80 160 mA and not 70 mA for the chip.
One slightly mitigating factor might be that as the multiplexing switches to different rows of LEDs some outputs will not necessarily be driven high all the time, so if a "cooling off" period was required the chip would get that.
However previous discussions have indicated that you are not supposed to exceed maximum levels, even momentarily.
One more question, from the datasheet from reply #4, what does it mean ±70 mA? What is the difference between that and 70 mA, since the chip won't be outputting negative voltages?
(edit) Corrected error. 8 x 20 mA is 160, not 80.