However, I think you are confusing voltage and current. The maximum output on pins QA-QH is ± 35 mA, but the total amount of current per device is ± 70 mA. So you will either need a transistor on each pin or consider using TPIC6B595N.
TPIC6B595 only sinks current. Use a resistor from 5V to supply current.Here's an example where the LEDs are wired up as if they were 8 LEDs in a common anode package.