Can a M74HC595 shift register directly drive LEDs?
Leds are rated for a given Current for ""full"" brightness for a specified life time. Most "regular" leds are 20mA typical, for x number of thousand hours. They can be driven at a lower current for extended life, or at a higher current at the cost of life (There are maximums to how much current can be used, as the current causes heat which then kills the led).

You can, and often should drive them with less current, unless you need full brightness. Just use Ohm's law to get the correct resistor for the 6mA current.

(Voltage Source - LED Forward Voltage) / Current (In Amps, so 0.006).

For the most part, leds at full brightness can be obnoxiously bright at straight angles. I've gone in to change the resistor for some gameboys because those things will blind you. 6mA is plainly visible in room lighting.