On my arrival as a permanent resident/becoming a citizen I saw how beautiful and clean Australia was, how maintained the river walkways, beaches, free sports and swimming facilities are from the Aussie tax $. I made the personal choice to pay tax on all my income to Australia. I also discovered in doing so there was a tax saving and less fees, reducing how complicated it would be for my tax agent to do my assessment.
If you are from the UK, see here for more detailed info on differences in having ATO not HMRC collect your taxes.
Should you also want to just pay tax in Australia, completing Form DT Individual (P85) is the way to achieve this.
ATO have data sharing agreements with overseas banks. Feel free to do a web search for yourself. If you are considering trying to hide or not declare your foreign income to dodge tax you risk serious repercussions if discovered.
I know ways to give less to the ATO and recover the money I do give them using their own rules. They are government initiatives because its win-win. The more you have in your retirement phase and ability to pay for private health cover, the less you need welfare or state funded medical.