Loot sharing system: how does the game assign loot to players

If I understand it properly, I think there's a problem in the question itself. As you said:

The merchants give you Scroll Fragments for white items, so we can sort of divine that that is the base item value. But it doesn't really flow well from there [...]

It doesn't add up, indeed, and you gave the exact reason in your own question:

• Path of Exile has done away with a single unified currency entirely. Exact
• [...] there's no way to tell what anything is worth, or which items are worth more as part of the local economy. More or less correct

Actually, while there isn't any curve to express the value of the items in terms of value, we probably could figure one out based on the difficulty to obtain the item from the vendors.

• You can get a Scroll Fragment with pretty much any crap item (which are easy to find).
• You can make a Scroll of Wisdom with 5 Scroll Fragments (which are easy to find, but you need 5 of those).
• You can get a Portal Scroll with Scroll Fragments (but you need 15 of those). It's getting harder to obtain, but still feasible.
• You can get an Armorer's Scrap with an piece of armour with a 20% quality, or with several pieces of armor with a total of 40% quality: feasible, but you'll need patience.
• You can get a Chromatic Orb with any item that has 3 sockets of different colors and linked together: you'll have to be very patient to find one of these.
• You can get an Orb of Chance with 2 rare items that share the exact same name: it's getting super hard.
• Etc.

With that in mind, the whole currency thing becomes a matter of recipes, and the "value" of each currency item is determined by its difficulty to be "cooked" (let's put it this way) and, therefore, its rarity.

The vendor recipe system allows the player to sell items to any town vendor in exchange for a multitude of currency items and equipment. Each recipe requires semi-specific items or combinations of items be put into the sell window at the same time, and the outcome will change based on any recipes that have been matched.

Source and list of known recipes: http://pathofexile.gamepedia.com/Vendor_recipe_system

The only remaining uncertainty in this system is the drop rate of each currency item (because, of course, it's also possible to obtain them by killing mobs). But I couldn't find any information about it.

The dev diary given by Gnoupi is also an excellent reading: Dev Diary: Rethinking Gold as a Currency.

There are four ways to find this out. You can remove all equipment and count your stats.

The second way is by visiting the bandits, starting with the one you most likely helped.

If they have an exclamation mark above their head it means you killed them, if not then they are the bandit you helped. In my case I helped Oak, as the pictures show. You can also re-kill the ones you previously killed for drops if you want.

Another more direct way is to talk with Eramir. One of the conversation options will be about the person you helped.

EDIT:

The fourth and easiest way is to look at your quest log.