Well, one sub-question at a time.

Suppose you are exerting a force of 300 N, how much KE is going to the train and how much is wasted to the ground?

Applying a force to an object doesn't necessarily involve energy expenditure. If I have a weight lying on the ground, then the weight is exerting force on the ground, but the weight isn't expending any energy by doing so.

Energy is spent by pushing on an object in the direction that it is moving, and the amount of energy spent is equal to the force exerted on the object times the distance that it moves. (If the force and the movement are in slightly different directions, ignore the part of the force that's perpendicular to the direction of movement, and just pay attention to the part of the force that's going in the same direction.)

Since the ground doesn't move when you push on it, no energy is lost to the ground. All of the energy goes into the train.

All other factors being equal, how does push-on-the ground affect the distribution of force/KE as compared to pull-on-the rope?

Well, you're asking about two different things: the distribution of force, and the distribution of energy. As I said above, all of the energy goes into the train, not the ground. How about the force?

It's probably safe to assume that while you're pulling a train, you're not accelerating at a significant rate (unless you're extremely strong or the train is extremely light). By one of Newton's laws, the net force exerted on an object is equal to its mass times its acceleration. Since you're not accelerating at a significant rate, the net force exerted on you must be approximately zero.

Of course, you are, in fact, exerting a lot of force, both on the train and on the ground. The reason that these forces can add up to zero is that the two forces are in opposite directions: if you're pulling eastward on the train, you must be pushing westward on the ground.

This explains why, if the forces suddenly become unequal (perhaps because the rope breaks, or your feet slip, or the train starts moving under its own power), you suddenly accelerate!

Which technique is more effective/favourable, which makes the train moves with less energy spent?

Since all of the energy goes into the train, which technique you use is pretty much irrelevant to energy expenditure. The kinetic energy of the train is determined completely by how fast it is moving. This means that, if no energy is lost to friction or other effects, then the amount of energy you must expend is determined completely by the speed of the train. (So if you want to move the train using as little energy as possible, the solution is to move it slowly!)

Now, there's kind of an implicit question here:

Which technique exerts the greatest amount of force on the train?

You'll definitely want to use your legs instead of your arms. In theory, the maximum amount of force you can exert on the train is equal to the maximum amount of force your legs are capable of producing.

There's a difficulty, here, though, which is that if a force is borne by a chain of components, all of the components are subjected to all of the force. If you have a rope in each hand and you're pulling with a force of 300 N, then your legs are exerting a force of 300 N and your arms are exerting a force of 300 N, and your torso is exerting a force of 300 N as well. Your arms will handle it all right, since they aren't moving; human muscles are better at holding still than they are at pulling something.

Still, it takes some effort just to hold the ropes with your arms. So it'll be best to remove your arms from the situation, by wearing a harness.

And there's one final problem. Suppose you're standing perfectly straight, and you pull on the rope while bracing with your legs. Since these two forces are applied to different locations on your body, the effect is that you'll rotate and fall forwards! In order to exert a large amount of force without falling over, you'll need to lean way back, exactly as the people in your pictures are doing.