I would like to know if the usage of the term is correct?
Don't worry, the rumour will fizzle out before you know it.
Could you give me other examples of its usage?
I wonder if it's interchangeable with the verb 'peter'?
It has two meanings:
1 make a feeble hissing or spluttering sound.
The light rain fizzled on the bonfire.
When we lit the rockets, they fizzled before they took off.
2 (fizzle out) end or fail in a weak or disappointing way.
The campaign fizzled out after the candidate admitted his errors.
You can replace "fizzled out" with "petered out" in the second usage.