Do ogres and other large creatures need to use weapons with the two-handed property with two hands? As far as i can tell, the rule is for player characters, and as there is no playable race that is large (that I'm aware of), the subject wasn't brought up, but this has been bothering me for a while. On a side note, if they do ignore the property, would a half ogre have this same ability by this ruling?
The specific rule for two-handed weapons can be found on page 147 of the PHB:
Two-handed. This weapon requires two hands to use.
Nowhere does it have any additional rules for creature size.
Does Enlarged make a difference?
There may be some guidance in the Enlarge/Reduce Spell (PHB, 237) with my emphasis:
This growth increases its size by one category -- from Medium to Large, for example. If there isn't enough room for the target to double its size, the creature or object attains the maximum possible size in the space available.
Until the spell ends, the target also has advantage on Strength checks and Strength saving throws. The target's weapons also grow to match its new size. While these weapons are enlarged, the target's attacks with them deal 1d4 extra damage.
While it doesn't state that the now larger creature can hold a two-handed weapon in one hand, it does give extra damage for the extra large weapon. **The text suggests that it is the extra size of the weapon that deals the additional 1d4, and not that it is an extra large creature wielding it **.
The extra damage delivered by the larger creature is due to the larger weapon wielded by the larger creature - not just the larger creature.
What this means for you: Size Doesn't Matter
As always, a DM can rule how they'd like, but it is fairly clear that the two-handed property limitation is not based on creature size, but on weapon property requirements. What matters is having both a larger creature AND a larger weapon for there to be an increase in damage or abilities.
Why just one-handed?
Andras' answer on is there a way for a medium creature to wield a two-handed weapon in one hand brings up a very valid point. If you're only doing this for fluff reasons and not as a means to dual-wield or have a shield, then there really is very little harm in doing so.
However, if he does not want Duelling or Dual Wielding benefits, and he is ok with keeping the other hand empty for "balancing", I see no problem allowing it.
At that point it has no influence on game mechanics, it is just aesthetics.
For NPCs, it's all about the stat-block
If it doesn't state that they can do something that isn't normally allowed, then they probably can't. Larger weapons with larger and stronger creatures generally deal more damage then the PC equivalent of medium creatures with their associated sized weapons - but unless it says they can do something that would be an exception to the rules, then they can't (like use smaller weapons from other creatures differently then what the weapon properties state).
Weapons and their sizes are associated with the same creature's size.There isn't a mixing and matching of weapon size and creature size to increase capability.
Let's say you allowed the large creature to use smaller creature's two-handed weapons in one hand. Just as a heavy weapon wielded by a small creature is done at disadvantage because it wasn't made for that size, a similar rule may be used in this case. The smaller weapon isn't made for the larger hand, and disadvantage may be used here as a balance. For every plus, there is often a minus to balance - this would be a possible resolution to the allowance.
Although not spelled out it is perfectly logical to allow this
Assuming that the situation is that a Large creature is attempting to wield a two-handed weapon sized for a Medium creature then, although the rules may not explicitly say so, logic and verisimilitude demand that the answer is yes. They should absolutely treat the weapon as if it were lighter. Anything else is patently silly. The rules specifically state that Small creatures get disadvantage on weapons with the Heavy tag, and a Tiny creature may not be able to use weapons with the Heavy tag at all, so there is a precedent in the rules that size can matter, but in the end it leaves it partially up to the DM (in the case of Tiny Creatures).
I believe that when the rules defy logic it is probably an oversight and it is up to the DM to correct it in order to facilitate the verisimilitude of the game. If Small creatures have disadvantage with Heavy weapons, why not let a Large creature treat a Greatsword as if it were a Longsword? Two-Handed becomes Versatile, and Heavy is unimportant for a Large creature. Do you think that a Longsword would fit two large sized hands on the hilt? I think it probably wouldn't, so I would remove the Versatile tag from a Longsword wielded by a Large Creature too. Again, this is assuming that the Large creature is using weapons sized for Medium creatures.
Large monsters set a precedent of changing normal weapon rules in a similar way
If you look at some of the monster weapons in the MM for Large creatures you will find that they are oversized versions of weapons that are not found in the PC's weapon charts.
Let's take the Minotaur listed in the MM on page 223. This is a Large creature that is presented using a Greataxe that deals 2d12 damage! In the PHB, a Greataxe is shown as dealing only 1d12. Obviously the Minotaur's larger size allows it to wield larger weapons, so it is only logical to assume that a Large creature could treat any weapon sized for a Medium creature as if it were one step lighter than it actually is. This is not spelled out in the rules exactly, but it is perfectly reasonable to allow.
Precedent with Enlarge Spell
When a Medium creature is Enlarged as per the spell, the rules are clearly spelled out. The weapon grows with the character and the damage increases by +1d4, which effectively changes a one-handed longsword into a weapon that deals 1d8+1d4[2-12 damage], which is essentially the same as wielding a great sword in one hand that deals 1d12[1-12 damage] but slightly worse. With that established, there really is no reason not to let a Large creature wield a two-handed weapon sized for a Medium creature in only one hand... unless the weapon does not make sense to wield one-handed no matter your size, but if this question was about longbows instead of great swords I would be extremely surprised! :)
Your DM will be the final authority
All that being said, if you present this case to a DM that disagrees, the DM is right and you should move on. It is no use getting hung up on something like this and letting it ruin your fun. If you are already being allowed to play a Large character, then you need to realize that the DM will need to take extra care to keep power levels balanced, so if they say no, that may be the impetus behind it. Remember, for it to be acceptable, it also has to increase fun for everyone involved.
The two-handed feature is specific to the weapon. If the creature used a smaller creatures weapon, it would likely be treated as a hand axe or similar, at the DM's discretion (but likely would not maintain the same damage stats). The weapons stats are a representation of how it is wielded more so than inherent damage value (as represented by Versatile weapons)
As NautArch mentioned, the Enlarge/Reduce spell scales items to the new size and adds 1d4 damage to represent this for weapon attacks, additionally the following two examples should help.
The Duergar special ability Enlarge turns them into a Large creature, and items scale to their size. They now deal double damage dice for strength based weapon attacks (1d8+mod -> 2d8+mod). However the Player Character Duergar version follows the rules of the Enlarge/Reduce spell.
Also in the official module Forge of Fury (new 5e version in the Tales from the Yawning Portal) There is an Ogre that uses a Battle Axe instead of a greatclub, and it follows the same rules as an Enlarged Duergar (double damage dice)
Unless it specifically says otherwise, I would say you treat a two-handed weapon as relative to the size of the creature wielding it. You may need to modify this weapon if A PC loots it before it can be used.