Renouncing Australian citizenship when becoming a Spanish citizen

Hi,
I was born in Australia and am a citizen, however as my father is Spanish, I'm considering becoming a Spanish citizen.
On the consulate's webpage, it says:

I'm wondering if this means only Spain would recognise this, or if Australia would recognise this as well?

AI insists that:

• Spain does not check with Australia
• Australia does not enforce Spain’s renunciation
• Thousands of people hold both passports without issues

Does anyone know if Australia some how checks with Spain or if there is some international register of people's citizenships?