I was looking around the net and there is awful lot of different answers, how much current are the chips actually able to provide.
The datasheet clearly says max. 35mA/pin, max 70mA through Vcc or Vss and power dissipation of 500mW
Yep. There's no confusion over it or "different answers". The specification says 70mA per chip.
I see a lot of projects around, that are exceeding these limits.
There's also millions of LEDs attached to Arduino pins without resistors, that doesn't make it right. Most of those people have never read a datasheet in their lives.
For sinking current, use a TLC5916 or TLC5940. They're current controlled and the difference in price is made up for by not having to mess around with resistors, etc. (IMHO)
For sourcing there's the TPIC6B595, but it's usually better to sink.