I dusted off my old copies of Pokémon Diamond, Platinum, and SoulSilver to test this. When you enter a trade in the Union Room, every Rotom in your party turns into a normal-forme Rotom, even if it isn't being traded to another game. The appliances get returned to the room in each respective game. I suppose Gamefreak anticipated this issue and implemented this to make sure people wouldn't run into any same Rotom-forme problems.
It's not possible to have two Rotom of the same forme. When you choose an appliance to turn Rotom into a specific forme, the appliance then disappears (until choosing a different forme), so there wouldn't be a way to have two of the same forme in your game without cheating.