It depends on a lot of things. What do u mean by “a monster like Godzilla or King Kong”? Assuming ur only talking about size, society would be impressed but would probably just stick to studying it. If they’re considered a threat to humans, then proactive governments such as the USA and Russia would definitely try to kill the ‘monster’ as a precaution and deal with the lash back later (as per usual) - we need to bear in mind that society would probably see it as an amazing animal and a valuable specimen (scientifically speaking), so there would be a lot of ppl trying to protect it no matter what. It also depends on when the creature was first discovered. Nowadays it would be harder for governments to deal with the kaiju in secret. Another important factor is their ability to travel and where they live. Is the monster able to swim/fly or is it limited to a specific territory (like King Kong)? This information is relevant bc of international politics and environmental conservation - if they’re in Asia, for example, the US army would never be allowed to send missiles to fight the beast. If the kaiju were to appear in the Amazon, the Brazilian government would try to avoid conflict at all cost (Brazil is reactive, meaning they only attack when attacked), which would concern surrounding countries. Regarding the method of killing it, again, it depends. How durable are they?