"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.