Open hands make trading almost irrelevant. Closed hands means trades could still happen but probably shouldn't often, so maybe it's not worth the complexity to allow trades. For me though almost all the fun of Settlers is in trading - trying to become one of the partners in as many of the inevitable trades as possible, even if the other player gets a little more than you every time, because you're still gaining on the non-trading players. The backdrop is the bits of tactics and strategy on the board, but other games are far richer in that department.
So how could we retain interesting trading dynamics? I'm thinking somehow add fake players, supply, demand to the mix. How about this (completely untested):
After the board is laid out, before determining first player, jointly agree on a ranking of resources from 5 to 1, most important to least, probably similar to "least pips available to most pips available". For example, maybe it's wheat > ore > brick > sheep > wood. I'll use that ranking for illustration.
Create a demand deck of 2x wheat, 2x ore, 1x brick, 1x sheep, facedown, shuffled, with a corresponding empty discard deck. Similarly create supply: 2x wood, 2x sheep, 1x brick, 1x ore. Deal 1 demand and 1 supply faceup. This is an available trade. Between turns anyone may discard cards matching the shown demand from their hand to gain cards matching the shown supply, once.
Details: Player who just took a turn has priority. If one of you is about to trade, the other may outbid them, repeat until a highest bid is reached, then that trade goes through; discard all the demand and supply cards to their respective discard piles. If no one makes a trade, deal an additional demand and two additional supply; the available trades are "the first N demand for the first 2N-1 supply", higher N wins. If a turn ends and there is no available trade, deal one. Whenever an available trade has wlog a sheep in demand and in supply, discard that match. If either the available supply or demand has no showing cards, discard all showing cards. Whenever resources (non-7) are rolled, add a resource of each type among the most distributed to the supply discard pile and a resource of each type among the least distributed to the demand discard pile. Whenever a supply or demand pile is empty and you need to draw, shuffle the discard, deal 6 facedown to form the new deck, and return the others to the bank. If there still are not enough cards, discard the available trade. When you play a Monopoly you also gain any matching cards in showing supply. You may not steal from the Shadow Trader.
Outbidding: only a superset of the showing demand is a valid bid. Higher N wins. Values are 4, 3, 3, 3, 2 in order of resource rank; highest sum of values wins. If an extra supply card exists, you may decline it and add its value to your total.
Example of play: there's hardly any wheat or ore, lots of sheep and wood, so above ranking is used. Showing trade is demand: wheat, supply: wood. Surprisingly, lots of wheat has been placed on and rolled. Alice offers a wheat. Bob offers a wheat and a sheep. Alice folds, Bob trades. Bob takes his turn; 2 wood and 1 brick are distributed, so 1 wood is added to supply discard and 1 wheat, ore, and sheep to demand discard. A trade is dealt, demand: sheep, supply: ore. Both players are chagrined to not have sheep, and do not trade. Alice takes her turn, rolling a 7. Additional trade cards are dealt, demand: sheep / wheat, supply: ore / sheep, wood. Discard the sheep. Demand: wheat, supply: ore / wood. Alice offers a wheat. Bob offers a wheat, declining the wood. Alice offers two wheat. Bob folds, Alice trades.
Here's a way to play Settlers of Catan with two players that David and I worked out this weekend. We've only played one game with it so far, and didn't think very hard about tradeoffs, so there are probably better ways to do it.
Set up the board as normal.
Each player controls two colors. We did white/blue and red/orange.
Placement is A B B' A' A' B' B A
Turns aren't color specific, except that you can only build for one color on a turn.
If you own a port with either color you can fully use it on your turn.
Development cards are color specific, which really only matters for the 1 VP cards
Each player has one hand, and the robber limit is 14 instead of 7. Possibly this should be more like ten or even seven.
The game still ends when one color gets to 10 VP, but the goal is to not come in last. Whichever player has the color with the fewest VP at the end of the game loses. The player who got to 10 VP first wins ties.
Trading is allowed, I guess, though since the game is zero-sum you wouldn't actually trade with each other.
Hands are open, not to facilitate trading but to make the strategy more interesting by letting you better model your opponent.
We played two games, but stopped the first one very early on. We agreed that one of us had an 80% chance of winning and so counted it as 80% of a win. Like a doubling cube this helps concentrate your time onto interesting games.
I enjoyed playing, though I don't know how many more games of it I'd like to play. In generally I enjoy the dynamic of games with more people a lot more, though under the current circumstances that's not always practical. It has less waiting than regular Catan, which is a plus.
I think the 36-card variant ("gamblers fallacy dice") would help here a lot more than in regular Catan, since otherwise the game can get unbalanced quickly due to quirks of the die. In general I like the way the card variant adds strategy, especially if it's easy for the players to tell what cards are yet to come.
Because there's no trading, ports are a lot more important than in a regular game. David picked a port-heavy strategy while I went for variety, and (n=1) ports works better.
After playing I looked around for other sets of two-player rules and there are a lot of variants out there! I haven't looked through them in detail to figure out whether I like this one best.