Trading NEEDS to happen, in some way, shape, or form. In my opinion, there should be private trades (like what Christopher Cannonskull mentioned early on), public auctions, and gifting.
For Private Trades, well, pretty simple. You can view each other's inventory and you can do several things. One of them is that you can put one of your items in the trade box (or a heap of gold, whichever one). The second is that you can click on something in your friend's inventory and make a request for this. The person can then say what they want for this. It would be complicated and there would need to be some good haggling features, but it should be able to work out.
For auctions, you would be able to broadcast out to your entire server what you are selling, and what you want for it. There would be an option in the sea chest for this. There would again, be several buttons to click. You would choose the item (auctioning is one item at a time) you are selling, how long the auction will go for, if you will accept items for bids (these bids will be placed in a different category), and starting and buyout prices (this does not have to be gold). Should someone bid an item along with their gold, the player will get a notification, and they can decline or accept that one on the spot. Once the auction time is up or someone bids buyout price, then a notification will go to the person who started the auction giving them the highest bidder, and they can finalize the trade. Also, they should be able to refuse service to anyone. There are some people you just don't want to do business with, aye?
This somewhat goes along with auctioning, but you should also be able to broadcast if you are looking for a certain item, and you can put what you will trade for it. Let's say that I want a Behemoth Blade (I do...). I would go to a spot in my sea chest (Now that I think of it, it could be placed in the Lookout), and I could choose from a list of weapons (no typing... you've seen the horrible spelling that some have.) what weapon I am looking for, and what I am offering for it. If I get any people willing to give me a Behemoth Blade, then I get a notification saying so, and I can finalize the trade.
Then of course, gifting. Let me first say that it should be just between friends, guild, and crew. I don't want to have level 5's coming up to me offering me ten gold and a rusty cutlass in hopes that it will make me want to be their friend, because I'm sure that would happen. As an option on the player card, you could click on a little icon looking like a present or something, and then you could pick something from your inventory to give to them. A message pops up on the other person's screen (similar to a friend/crew request) saying "[Pirate Name] would like to give you [Item Name] as a gift. Do you accept? [Yes] [No]
Anyway, I could probably type more, but I won't bore you. That's my opinion on trading. It really needs to happen, but POTCO would have to take MAJOR measures to make sure that it doesn't get abused. Maybe they could limit you to X amount of trades a month, or something. Just something so that it doesn't get abused, because I agree with you all, it is something that can change the game's economy completely, for better or for worse.