I'm ESL, hope I can deliver the questions clearly.
Consider a regular and sharp longsword.
Would it be possible to stab someone by handling it close to the tip or maybe in the middle of the blade? (I never had one in my hands, I have no idea if you have enough stability to stab).
How much damage would it do to the hand?
The answer is Yes, of course, a warrior can grasp his own sword by the blade, and the old fighting manuals show this technique clearly (half-sword technique). There are also instances where it is recommended to grasp the opponent's sword.
Here is good treatise on the topic of half-sword technique, when it's used & why.
From Medieval Combat (a translation & description of Talhoffer's Fechbuch), we read: "The half=sword techniques were originally devised for fighting armoured opponents. The sword is gripped on the blade with the left hand and used like a short spear or bayonet to slice or stab into the armpits, groin, face, throat and joints."
Even though the technique was devised for armoured fighting, it could certainly be used for fighting without armour. Apropos to the question of injury to the swordsman's hand, note that medieval guantlets did not fully encase the hand and fingers:
The swordsman still had to wear a (thin) leathern glove underneath. This does not invalidate the answer, because the leather itself is relatively thin and offers little protection from stabs and slices. Thin leather = less bunching & greater range of motion; thick leather = bunching & less range of motion. You still need to take care in order not to slice your hand through the glove!
Zweihander Techniques Demonstration
Yes, it's possible to stab someone by holding the middle of the blade.
As a weapon, the long sword was often used with half-swording techniques; so often, the other hand would be holding the blade.
Half Swording was a thing back in the day, where people would grab the middle of their blade with their hands while using the sword. The sharpest part of a sword is often closer to the tip, so this was relatively safe as long as you were wearing some sort of hand protection (eg: Plate gauntlet/ Thick leather gloves/ etc.). Even without the hand protection, it is theoretically possible to grip the blade such that your palm and fingers would not be sliced open.
This actually allowed the sword to be "shortened" for more powerful thrusts, or have more leverage when in close quarters.
Also often done was the flipping around of the sword to use the handle side as a mace/war hammer type implement.
With a sharp-edged longsword (no ricasso , no gauntlets and no cheating by holding sword's hilt with the other hand), not so much damage.
Skin will be punctured, and an eye can be lost, but there can be no deep penetration, because the attacker's hand's grip will slip over the sharp edge. That hand can be cut to the bone, if attacker is trying hard.
Two techniques above would be sufficient to kill an unarmored opponent, but still not likely to be effective even against a light (like leather) armor.
I've actually had this problem in real life before. Couldn't explain it well without drawing it.
I pushed the butt of my hands together along the face of the blade while locking fingers over the blunt end of the blade.
Of course this method won't work with 2-sided blades
I used this method as I had snapped the hilt of my sword off
The other answers discuss the technique quite well, so I wanted to toss this in.
According to http://www.darksword-armory.com/how-sharp-were-medieval-swords/, The relative sharpness of the longsword varied across time. If the blade was destined to attack an unarmored or leather clad warrior, it's edge would have been sharper (also more brittle). In this era, we would assume half-sword techniques without any sort of hand protection to be potentially catastrophic to the hand.
However, with the advent of chain mail and plate armor, razor sharp edges became less desirable. Less razor-like equals less damage to the bare hand.
Depending on your time period (and ultimate sword design), your character could lose a finger or two from the trauma. On the other hand, the sword may not even be sharpened up to a point, if it's intended target is plate-clad. Minimal/no damage could be possible.
Yes it is possible, it's called half-swording and it's quite common in a good amount of longsword manuals.
With a proper way of gripping the sword you can half-sword safely even without a pair of gloves and sustain no damage to your hands. Skallagrim has a video in which he does exactly that, half-swording with his bare hands then Mordhau-ing a rubber tire, although he admits that wearing gloves give a psychological advantage to half-swording.
You could do it with one hand on the blade and the other on the hilt (aka Half-swording), but doing with with both hands on the blade would not work. You would not have enough grip to hold the blade firmly and it would just slip off your hands. That's why cross-guards were made, so you can have a point of contact when you thrust. That part gives superior grip and prevents the hand from slipping on the blade.
Even if you had a good grip, the damage would be superficial, you would only "scratch" your opponent, no serious injury. In fact, you might hurt yourself more than you hurt your opponent if you lack hand protection.
