Here are some educated guesses from a physicist about what would happen if the Earth were split in two. I'm assuming that the splitting happens magically and instantaneously, with one half just suddenly disappearing and reappearing somewhere else. Although this leads to quite extreme effects it's actually the best-case scenario - any other way of splitting the planet would add energy to the system, making it even worse.
I'm also assuming the split happens along the Grenwich Meridian, so that it splits England, Africa, both poles and the Pacific Ocean into two, while the Americas and East Asia end up in the middle of their own hemispheres.
For someone in central America or Indonesia, the very first thing they'll notice is that gravity is suddenly half of its normal strength. This will probably feel like a terrible earthquake, as if the ground has suddenly started to drop away underneath them. This is just an illusion, though - it hasn't. Yet.
For someone closer to the edge of the hemisphere, gravity has not only reduced but also changed direction slightly, since it now points towards the centre of gravity of the hemisphere instead of to what used to be the centre of the Earth. However, this is the least of their problems and they will not have much time to think about it, because of the enormous shockwave that's passing over the land at the speed of sound, destroying everything. This occurs because the air at the edge of the hemisphere is suddenly in contact with a vacuum and is rushing into it. The air that goes over the edge will then start falling towards the core of the Earth.
Speaking of the core, this has now exploded. It was previously under a huge amount of pressure, so much so that the inner core is solid iron, despite being hotter than the outer core, which is liquid. Once that pressure is released it will turn back into a liquid, expanding as it does so. I'm not sure exactly what the volume change is, but I'm sure there are now unimaginable amounts of white hot liquid iron rushing out into space at 6000C. This explosion can't be seen from the surface, but it causes a shockwave of its own, which will travel up through the mantle and manifest as a huge earthquake, which will destroy everything the atmospheric shockwave doesn't get. For places far from the edge, the earthquake will arrive before the atmospheric shockwave, because it has less distance to travel through the Earth.
After that, the half-Earth will start collapse under its own gravity to turn back into a sphere half the size of the original. The Earth's core is now entirely liquid and the mantle, while largely solid, doesn't have anywhere near enough strength to hold itself in a hemisphere shape without collapsing. I don't know how long this collapse will take but it will be well underway within hours if not before. If you watch simulations of the Moon-forming impact you'll see that the Earth behaves like a liquid when it is subject to forces on that scale.
The process of collapse turns gravitational potential energy into heat, with the result that once it's finished the half-Earth will have a surface of molten rock and an atmosphere made mostly of hot gases that were previously trapped in the mantle and core.
In short, most people will be dead within hours, and long-term survival is completely impossible.