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Employing the services of an enchanter is one of the trickiest transactions you may attempt to perform with other players in the world of Azeroth.
Why is it so complicated?
As a player you will find or buy weapons or armor for their primary features as they benefit your class. For example your low to mid-level hunter may collect a set of armor items that are "of the falcon" (giving bonuses to agility and intelligence) and your rogue would typically favor items that provide agility and possibly stamina (perhaps "of the tiger"). These items will normally be soulbound so they can't be traded, sold, mailed to other characters nor stored in a guild bank.
Players often wish to place additional enchantments on these items to further improve their usefulness. That's the job of an enchanter. Because most such items are soulbound the enchantments must normally be performed in live "trade" sessions. (The recent addition of inscription vellums to the game allows enchanters to craft scrolls which can be sold, mailed around (including using COD) and traded normally. These can be used by any player to enchant their items just as armor kits have traditionally been used. However, the "live trade session" is still the most common way for enchantments to be performed in the game).
Enchantments require materials, often simply called mats. These are special "dusts" and "essences" and "shards" which have been extracted from other magic items. The enchanter has a "disenchanting skill" which he or she uses on magic items which he or she buys or loots during play. Those materials are frequently sold in the auction house and are the primary cost of getting any enchantment done. (They also make the enchanter's job of developing higher levels of his or her skill very expensive).
The Basics: How it Works
In the simplest case you are getting an enchantment with the materials included. You find the enchanter, getting close enough that you can open a trade. You place the target item into the "will not be traded" slot, (see: trade, step 3, second bullet point) and the agreed upon sum in the money boxes (as per trade, step 4).
He or she places the enchantment on the item ... and it turns green in your dialog slot. You can mouse over the item to confirm that it was the enchantment you agreed to. If so then you click trade; he or she confirms and your done.
Conceptually that is as simple as having a rogue pick open the lock on some box that you've picked up.
What typically complicates things are finding the desired enchantment and handling the materials. When you rely on the enchanter to have the materials then you are likely to pay more for them. There are a huge variety of enchanting materials and it's easy for an enchanter to fill up two 20-slot enchanting bags with common materials and yet not have the ones you need for the particular enchantment you want. Also the most powerful and popular enchantments have materials that are rare, and in high demand. It's very expensive for an enchanter to keep a stock of those materials on hand ... and they will charge a premium for the hassles they went through finding, acquiring and storing them.
Obviously you may be able to get a better deal if you can provide the materials. Often you can find an enchanter who is trying to "level up" his or her skill and will perform the enchantment for free. Sometimes you'll find one who is willing to throw in the materials for specific enchantments for free just so they can level up. (However, they are far more likely to just do that to one of their own items repeatedly ... or buy some vellum so they can sell the resulting scrolls in the auction house).
When enchanters offer "free" services it usually only applies to those spells which appear in "orange" or "yellow" in their craft link. The ones which are likely to help them boost their skill level. You should fully expect that they will demand payment for enchanting anything that is green or grey to them. They spent quite a bit of gold and effort to get those skills and this is their chance to recoup some of that back. Remember they get no benefit for doing grey or green enchantments for you. If they do one of those for free, consider it a favor! (Also you should expect to have to gather the materials for any of the grey/green enchantments that you request from them --- they will very likely be focusing on having materials only for the highest priority enchantments in their book).
Customer Supplied Materials
Anyone can buy or trade in enchanting materials. Starting by finding the desired enchantment and making a careful list of the exact materials which are required for it. Then find each of those (in the auction house or as loot or by trading with ther players). Be sure to collect the correct number of each.
Most of the Enchanting ingredients are dusts, essences, and shards which are extracted from other magic items through disenchanting or primal elements which are created by high level alchemists. However, some enchantments may require various herbs, ores, metals or other items.
To use them one must typically perform two trades with the enchanter. First to hand over the materials. Then again to perform the spell as already described. This is one of the few cases where a player is at risk of being duped by another player ... of having in-world valuables stolen. An unscrupulous player could offer to perform an enchantment with your "mats" ... accept the first trade and then run away or log off. (Pay attention to the characters name before you hand over your materials ... and report any such incident immediately via an in-game help request. I have no idea how the game masters would respond ... but it's your best hope if something like this happens).
Obviously it's best if you can simply do business with people you know ... with those you have played with or who are members of your guild, etc. I've never had anyone do such a thing and most of the enchantments where the materials are expensive enough to worry about are only done by high level characters --- who usually don't have to steal from other players. However the risk is worth mentioning when you have shelled out a couple hundred gold for the materials that are required for a particularly desirable enchantment.
One technique would be to have the enchanter trade a deposit (roughly the fair market value of the mats) on the first trade ... then you pay that back plus your fees and tips during the second trade. If either party runs off between the first and second trade than the loss is minimal (assuming the mats were purchased at near or below fair market prices and that the market for reselling them isn't glutted). One problem is that the enchanter may not have enough gold on hand to handle such a "deposit" (or may simply be unwilling to risk the inconvenience of being stuck with materials to be auctioned off). Another is that markets fluctuate. Agreeing on the "fair market value" of a given set of materials could involve some haggling.
In general this has not been a problem ... but forewarned is forearmed.
If the enchanter has the materials and you also have them, then you can easily and safely perform the trade ... your money (for his or her fees and tips) and your materials to replace those used by the spell. You may have benefited from getting a better bargain. Either way he or she has the same materials ready for the next customer and only has to maintain one set of them on hand.
(Ideally Blizzard could offer some sort of enhanced, three phase commit, transaction dialog ... but most users probably couldn't figure out how to use such a thing even if they offered it. Getting customers to deal with the simplest enchant/lockpick transactions is hard enough.