Seeds are tiny miracles of life all wrapped up in a hard protective coating. That coating has to be hydrated, or exposed to water, to soften and encourage the seeds to produce the enzymes necessary for germination. The seed contains enough food to nourish the plant-to-be through the germination and sprouting process. After the seeds have sprouted, they need warmth and light to continue to thrive. Splitting a seed to produce two plants instead of one isn't a good idea.
If the seed has sprouted and one half is still attached to the sprout, it can be planted and may grow, as long as the embryonic root and stem isn't broken in the transplanting process. This occasionally happens with an avocado seed for example.
Seed Cut in Half Vertically
The seed will not sprout.
Seed Cut in Half Horizontally
Nothing will happen because, in cutting the seed in half, you've also cut the baby plant-to-be in half as well.
Half Seed Turned into Organic Matter
Broken seeds can be added to the compost bin or planted around growing vegetables and flowers. They won't sprout, but they will add nutrients to the soil as they decay.
Half Seed Eaten
The half seed may be eaten by birds scratching through the dirt, by rats, chipmunks, or mice.