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:

The person must declare that they renounce their previous nationality.

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?