The first instance of a westerner being poisoned by Fugu I can find is a reference in Captain James Cook's log from his second Pacific voyage of 1772-1775. On the 7th of September, 1774, the crew of Cook's sloop HMS Resolution caught and ate the flesh of several ‘spiny tropical fish’. They fed the less desirable parts (the heads, organs, etc) to the herd of pigs that was kept on board. The crewmen who ate these fish fell ill, suffering from weakness, numbness and shortness of breath; but they all survived, suggesting a mild dose of toxin. The entire herd of pigs perished however, suggesting that they had eaten the livers and ovaries of the Fugu. It is the liver, eyes, and ovaries of the puffer fish which contain the highest concentration of tetrodotoxin, leading to a lethal dose.

However, people in the Western Pacific have certainly been eating different species of pufferfish for thousands of years. There have been finds of pufferfish bones in the rubbish middens of settlements dating to the Middle Jomon period of Japanese history (2500-1500 BC). This would be long before written language was common in the area, as we have no written records of Japan at all until the much later Yayoi period, with the King of Na seal from approx. 57 AD. That being the case, by the time anyone would have the tools to write down how to safely prepare fugu, the practice would already have been ancient. It's similar to how we have no records of the first domestication of wolves, or wild horses.

That being said, and I admit up front this is informed speculation, I imagine the first people to prepare and eat fugu followed the same sorts of methods used to test if a wild grown plant is safe to eat. It is common in both plants and animals for some parts to be poisonous while others are not, so you first separate the food item into it's constituent parts. In the case of a plant, you would separate the roots, the stems, the leaves, any fruit, seeds, etc. In the fugu, you separate the flesh from the different organs. You must then test each portion of the food item individually. You can begin by rubbing the item on your skin, to see if there is a contact reaction, as with poison ivy. After waiting a few hours, if there is no reaction, you proceed to the next step: touching the food to your lips, and tasting it with the tip of your tongue. You must do this for at least several minutes. If there is no negative reaction, you can place the item in your mouth without chewing. If after several minutes of holding it in your mouth there is still no reaction, you can then chew the item. If after chewing a small piece of the item for several more minutes you still notice no reaction, you can proceed to swallow it. Now, you should wait at least 8 hours to see if there are any further adverse effects, while consuming nothing else that might confound your results. If you were to do this with different pieces of fugu, you should be able to determine which are suitable for consumption and which are not. If you followed these steps with the fugu liver, you would likely never get to the swallowing stage, or even the chewing stage, as the high concentration of toxin would quickly numb your lips and tongue. It is by this slow process of trial and error that you can safely determine what is edible and what is poisonous.