Well there is a significant diffirence between simply being able to wield a d12 halberd and being cable to wield a d10 halberd (or d12 greataxe, or 2d6 maul) and use a heavy shield. I think that the latter is too broken and not the intent. With one you get essentially +1 average damage, with the other you get the use of a sheild and up to a +1.5 average damage (in comparison to one handed weapons). Again, too powerfull. Besides, if the intent was to allow ytou to wield a weapon as if you wee a large creature, it would say so, but it doesnt."You can use weapons of your size or one size larger than you as if they were your size"
You are medium, one size larger is large.
You can use [medium] or [large] weapons as if they were [medium]
All weapons in the PHB are medium, large weapons are easily found off the corpses of some large creatures, I assume they could also be specially made by a blacksmith of considerable skill. A large weapon uses a weapon die one higher than its medium variation according to page 220 of the PHB.
~~~~~Heres where it gets questionable~~~~
Naturally large creatures can use a medium two handed weapon as if it was a large one handed weapon. Since the idea of the Bugbear/Mino Oversized trait is it make your hands and arms "Large", It could be easily allowed that both classes could use 2 handed weapons in one hand. The average damage between using a large one handed weapon and using a medium two handed weapon with one hand is the same for most intents and purposes. One of the biggest issues that arises with this intepretation is the issue of reach, which I would allow as it is in no way game breaking.
Since the idea of the Bugbear/Mino Oversized trait is it make your hands and arms "Large", It could be easily allowed that both classes could use 2 handed weapons in one hand.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.