Should I use pull up resistors in this configuration?
For the outputs you configure as open-drain you'd need the pull-up. Not if you have push-pull outputs. Open-drain is typically used for wired-ORing outputs, you usually don't use them for a direct output-to-input connection. Push-pull outputs are faster because the low impedance allows for fast edges, while the asymmetrical output of an open-drain will give you slower positive edges.  

In the schematic of page 4 open-drains are used to allow different output voltages, depending on the voltage level the pull-up is connected to.