Below are the images of the definition of gutter on Google, which is a trough on the edge of a roof. However, in the phrase 'throw money in the gutter', I imagine gutter means something related to sewage.
When something goes into a gutter, it eventually goes down the drain, rendering it irretrievable.
1a : a trough along the eaves to catch and carry off rainwater
b : a low area (as at the edge of a street) to carry off surface water (as to a sewer)
c : a trough or groove to catch and direct something
It doesn’t much matter what kind of gutter or which drain. However, unless you’re a roof rat, you’re more likely to be engaging a metaphorical street gutter for your “throwdown”:
the edge of a road where rain flows away
In older times, without trash collection and good plumbing, it would be a filthy place full of rubbish.