Is the salsa verde meant to be made fresh or does it get better after a day or so?

I think it's best made fresh. The herbs lose their vibrant color and flavor if you make it in advance and refrigerate. If you are making the salsa verde in a food processor, it is literally a 10-second process so there's really no reason to make it ahead anyway. ;)

For this Italian Salsa I bet it's ideal to serve fresh at room temperature and wouldn't benefit from sitting in the fridge.

I'd say it gets different. I love the stuff and make it often in the summer - usually more than needed - to have around for a few days. It's good on so many things! Right after making, it tastes most fresh and cleanly herb-y, obviously. But it's still quite good for a couple of days, it's taste/texture is just more melded. It's pretty quick to do in a food processor, so I'd say to make it to serve immediately, but you'll be glad to have any leftovers!

One of the recipe commenters made it the day before and enjoyed it.

Knowing my own personal preferences for handling Italian parsley, I'd probably make it right before, not let it sit.