What is the relative value of 1 Coin in the implied setting of Dungeon World?
The game system of Dungeon World intentionally keeps the concept of money and wealth fairly abstract compared to other, more simulationist systems. The equipment section of the rules list costs in a manner clearly intended to only be taken in the context of "adventurers purchasing adventuring gear during an adventure", such as having simplified numbers of coins modified precisely by a character's Charisma score; it should probably not be taken to mean that all the NPCs use these rules themselves all day every day.
While this simplifies and expedites the bulk of the shopping that is most frequent in the game and slides over the stuff that is largely unimportant – as Dungeon World is excellent at doing – it leaves a logical void in the event that the story does take a turn that involves wealth or economics in some significant way.
How much does the typical Peasant Farmer or Commoner make in a month? If a DW group comes across 100 gold coins, is this a staggering "more wealth than a commoner has ever seen", or a more modest "a solid stack of money, but nothing outrageous"? What would be considered a "fortune", a "king's ransom", or "enough riches to retire in lavish style"?
How does 1 Coin in Dungeon World compare to 1 Gold Piece in Dungeons and Dragons? D&D being a popular game that does answer these questions and has a (deliberately) similar system, how do the currencies compare? A valuable listed item in making a comparison is that of food, which seems to be priced higher than D&D. However, weapons and armor seem to be priced much lower. It seems that things that commoners would need to worry about (food, shelter) are more expensive, while adventuring gear (weapons, armor, travel gear) is cheaper. I suspect that 1 DW Coin is, in general, worth less than a D&D gp, but probably more than a D&D sp.
Best Answer
Since you had several questions, I'll split my answer up a little.
How much does the typical Peasant Farmer or Commoner make in a month?
Well, a day of mundane labor is listed as 10 coins per week in the rulebook. One can assume that this usually means hauling heavy things about and so on, but I would say one could also rule it to be about the same for a farmer. Jobs that would require more education receive more, as indicated by a chirurgeons' healing being worth 5 coins. Given that feeding oneself each day for someone that actually lives in the current location wouldn't cost them 2 coins every time (after all, they buy ingredients, not 'luxury' meals at the Inn/Pub), that seems reasonable.
If a DW group comes across 100 gold coins, is this a staggering "more wealth than a commoner has ever seen", or a more modest "a solid stack of money, but nothing outrageous"? What would be considered a "fortune", a "king's ransom", or "enough riches to retire in lavish style"?
This is all down to estimations, really. I don't think a 100 coins are something that a commoner has never seen, but it is a lot of money. More than you would earn in a month as a farmer, but not entirely out of reach to save up for I'd say. Lets look at house prices! A cottage costs 500 coins - I'd say that should be within a laborers' reach if they have some skill, especially since a months' upkeep is 1% of the initial cost. A 2.500 coin House, however, would be more of an inheritance for a farmer, or self-built, obviously. Anyone rich enough to afford a mansion would probably easily have another 50.000 coins lying around for no reason whatsoever. Enough riches to retire in style? I would go for many hundred thousands, maybe even a few millions depending on how much style you want.
How does 1 Coin in Dungeon World compare to 1 Gold Piece in Dungeons and Dragons?
This is the core of your problem. It really doesn't. They are not comparable simply because in Dungeon World, things are priced based on how elaborate or unusual they are. Weapons, Armor, these are common, even daily things. A writing kit or a horse are more valuable than most armor! Staying at an Inn can be more expensive than weapons because its a luxury. A compass might be worth 50 coins even, as its more intricate and difficult to make. Even Magic Items as such seem more rare (and legendary/unique) than they do in D&D, therefore probably raising their value considerably.
In the end, you decide
If you don't like it, change it. Honestly, feel free to do so. As a group you create the world and with the world comes what things are valuable and which aren't... and why. Be thoughtful about it though, the value of things can really change the feel of a campaign (f.e. make it very low or high magic).
Don't get bogged down being too simulationist either; I would advise a little trick instead: Use X Gear. If there is adventuring gear, why can't someone get 2 uses of kitchen gear if they buy some, to have 'just the thing' in the moment. Or even bandit gear if they loot a group of bandits? Then you can use a slightly smaller (or higher) amount of coin compared to the Adventuring Gear cost if the group chooses to sell 'all that stuff'.