"Champions can use various weapon styles, but focus on using two weapons simultaneously."
This is not obvious from the champion skill set however because some key champion skills give additional damage depending on the main hand weapon damage per hit. An extreme case is Brutal strikes which gives damage of 3 x main weapon damage plus a bonus. Obviously a big slow two hander will do better here than two smaller single hand weapons. Turbine recognised this issue in patch 1.12 and tried to rebalance by reducing the fervor requirement for Savage Strikes and Brutal strikes by one point when dual wielding. I cannot say whether this has balanced the skill based damage between one and two handers but it does go some way towards addressing it.
A more basic question is whether or not the raw DPS of dual wield is more or less than the raw DPS of a big two hander. Typically dual wield weapons do less damage than an equivalent level one hander but more than half the damage. For example a lvl 8 white quality one hand short sword does 3.8 dps. A lvl 9 white quality two handed great sword does 4.6 dps. If there is no penalty for dual wielding then the double attacks from dual wield will do more damage than a two hander. Off hand penalties are common in RPG games but I could find no reference to one in the game literature or on the forums so I set about conducting my own experiment. Early tests with various weapon types indicated strongly that an offhand penalty does exist but it was possible that different weapon speeds and other variables were disguising the results so I decided to do a more rigorous test. I bought two identical Steel Banded Maces (Speed 2.3, DPS 2.8 (5-8 damage), Small Chance to Stun) and sent my level 9 dwarf champion to the western part of Thorin's Gate to observe the results of hitting level 5 Young Aurochs. I chose the Aurochs because they have no debuffs to interfere with results and I made sure to always pick level 5 targets. I chose the maces because they were cheap and they do not interfere with armour or hit chance.
I recorded a total of 29 single wield attacks and a total of 31 dual wield attacks (62 hits). The unadjusted average damage from the 29 single wield attacks was 7.7 and the unadjusted damage from the dual wield was 5.6 for main hand and 5.3 for off hand. However these results were strongly affected by random failures due to blocks, evades and misses. I do not believe that I have a large enough sample to allow for these so I adjusted the averages to take out the failed attacks. The average damage for successful attacks was 9.3 for the single wield main hand, 7.6 for dual wield main hand and 5.9 for dual wield off hand. These figures agree well with the most commonly observed hits in each case (the mode). The most commonly observed values were: Single Wield main hand: 9, Dual Wield Main Hand: 8, dual wield off hand: 6. Remember that identical weapons were used in main and off hand tests.
Although my sample is not particularly large it seems certain that there is a damage penalty for both main and off hands while dual wielding. The observed reduction in damage for successful attacks was 18% reduction for the main hand and 36% reduction for the off hand when dual wielding.
My tests suggest that there may also be a reduced chance to hit for dual wielding but I have do not have enough data to conclusively confirm this.
While conducting these tests I also noticed that single wield off hand does not appear to work. Even though the off hand weapon is visible in the fight animation the weapon seems to be ignored in the damage calculation and only fist damage is recorded.
Edit: A Poster to Codemaster's Lotro forums questioned the validity of my results in this piece. He or she was rather rude so I was going to ignore them but perhaps they were just having a bad day so instead I will give them the benefit of the doubt and clarify my results. Firstly please note that these results refer to auto-attack damage only. I don't think a statistical test needs to be done for skill damage because the skill description tells you the damage in both single and dual wield cases. My tests say nothing about failure rate (miss block etc) and a much larger sample would be required to be able to say something useful about this. The good news is that the standards error in each of my tests is between 2% and 3% so the conclusions with respect to damage reduction can be relied on to a high degree.