What training is required for pilots conducting ASETPA operations?
According to the CASA AOC Handbook, section 15.3.1.2, pilots do need to have specific training. Most of this training is regarding engine failures and emergency landings, in some situations in IMC.

CASA requires that all ASETPA operators have a training and checking
organisation.
The CASA Inspector will verify that for ASETPA
operator’s the following additional procedures are included within the
training and checking organisation:

Engine failure or malfunction which necessitates stopping the aircraft on the ground
Engine failure or malfunction which necessitates an off-airport landing after take- off on the most suitable terrain in the vicinity of the aerodrome. These can be organised into a number of procedures, depending on the height of the aircraft at the time of the failure
Engine failure or malfunction which necessitates turning to execute a glide landing upon a serviceable runway, including a ‘turn-back’ manoeuvre. This procedure is to contain a minimum indicated airspeeds and altitudes
Engine malfunction, during climb, cruise and descent and from the approach to land phase in VMC and IMC
Additional procedures for the conduct of a forced landing in IMC to ground level


The next section discusses route limitations, which I am assuming the pilots should at least be familiar with.