Purpose of series resistors in this bipolar rectifier circuit?
It's a series resistor for the zener diode. One is needed to make the zenerdiode able to cut off the exessive voltage. The zenerdiode draws just that current which is needed (in addition to the load current) to drop the voltage to the zener voltage. This is the way how zenerdiodes are used as voltage regulators.

This principle is practical only for low power systems due the continuous dissipation in the zenerdiodes.

The wattage of the resistor: Let Ux = the maximun DC voltage in capacitors after the rectifier, for example 20V. It's about 1.4 x the output voltage of the transformer.

Calculate the voltage drop in the series resistor: Ux - Uout. If Ux = 20V then the drop is 8V. NOTE the zenerdiode draws all the current that's needed for 8V drop in addition to the load current. The resistance is detemined in accordance with that.

The dissipation in the resistor = (Ux - Uout)^2/R = (8V)^2/270Ohm = 0.24W

Theoretically 0.25W resistor is ok, but in practice it easily gets hot due poor cooling. Take at least 0.5W, if Ux=20V happens to be your case.