I read some brain encyclopedia ages ago, so details are a bit foggy, but when the connections between the two hemispheres of the brain gets injured/damaged it can have some really strange effects. In the book they retold tests where they did things like showing a picture of a shovel to only one eye and ask the person to think of a garden tool, and they'd say shovel. Then they'd ask the person to explain why they're thinking of a shovel, and they'd tell some elaborate story like having met their neighbour a few days ago who said that they need to buy a shovel, or something similar. This due to the connection between the part(s) of the brain that constructs the story having severed connections to the parts the receives and processes the visual input of that eye. There was a few hundred pages of tests like these with wildly different effects depending on where the connections had been damaged.

Without injury the input should be shared just fine, but since the difference is so severe when the connections are damaged I guess (with no credentials or so to back it up) that there could be some hard-to-measure differences in thoughts formed depending on which eye is used.