Does having two Goblin Arsonist on the battlefield cause their ability to deal 2 damage when one dies?
No, it deals 1 damage
Whenever you see a card's own card name within the the rules text of a card, it isn't referring to all cards with that name. It is referring to itself. I believe at one point, the Oracle text replaced this self reference with ~this~ (or perhaps, the database allowed searching using "this"). You can think of the triggered ability in that way:
From the Comprehensive Rules:
If you were looking for abilities that would trigger multiple times, you need cards that trigger "Whenever a Goblin ...". These cards only care about the creature subtype in question. For example:
Two Shenanigans won't trigger off of themselves, because of the another, but both will trigger when another Goblin permanent is sent to the graveyard from the battlefield. (Note: Boggart Shenanigans will do 1 damage twice, which is not the same thing as to deal 2 damage once.)