Well, for TI's 74HC595, you've got:
±6-mA Output Drive at 5 V
Continuous current through VCC or GND ±70 mA
and
IO Continuous output current VO = 0 to VCC ±35 mA
and Note 1:
(1) Stresses beyond those listed under absolute maximum ratings may cause permanent damage to the device. These are stress ratings only, and functional operation of the device at these or any other conditions beyond those indicated under recommended operating conditions is not implied. Exposure to absolute-maximum-rated conditions for extended periods may affect device reliability.
and output voltage levels VOH and VOL spec'ed at:
VOH: QA?QH, IOH = ?6 mA
VOL: QA?QH, IOL = 6 mA
So you can try and drive at higher currents, voltage levels seen are not specified, and long term there's a good chance of part damage/failure.
If multiplexing, driving high from a shift register and sinking with a transistor, I would use a more capable part: cd74AC164, spec'ed for 24mA output drive
±24mA Output Drive Current
http://www.ti.com/lit/ds/symlink/cd74ac164.pdfDoesn't have dual stages, but you have NPN/N-channel as current sink to determine when current will flow, so I don't think that's an issue. Use the 2nd DS line as an inverted chip select line - data is shifted when the pin is high.