If you get an octopus in a shop, like Octopi 'R' Us or Ye Olde Tentaculate Shoppe, chance is it has already been tenderized. But if you fished it yourself, then you should know how to tenderize it.
And yes, it is necessary. Octopus, by itself, would be a fine material for building slingshots and probably shoes - not for eating. It is rubbery and leathery and not unlike a marine chewingum.
I suppose that you could solve the problem by pounding on it with a hammer or something similar in your own kitchen - but why ? You can do it on the seaside ! Attract attention ! Shock kids into vegetarianism ! Draw a crowd !
Octopus, pristine
The octopus at this point should be dead. If it is not dead, read Humane octopus killing. The tougher part of the octopus is where the tentacles join the head (they are thicker there), which means that they require the most abuse.
Find a hard, clean rock. It should not be covered with a thick mat of algae, and it should not be crumbly (unless you enjoy eating bits of rock).
Grab the octopus by its head (I am sure that if you are a squeamish person you should not be reading this), using the convenient side opening that Nature provided. Do this with the hand you would not use for throwing a ball.
Lift the octopus into the bracing marine air, and slowly drop it into your free hand. The octopus should, at this point, be sitting into your throwing hand with its head on top, looking at you in a reproachful way.
The gentle art of THUD
Now comes the fun bit.
Throw the octopus, with as much force as you can muster without pulling muscles, against the rock.
THUD ! the octopus will bounce a bit. Pick it up. Place in throwing hand. Repeat. THUD. Repeat. THUD. Repeat until the octopus turns a pale grey, and its tentacles hung limp.
Be careful with the throwing, because if you don't have all the tentacles gathered in your hand they will
whiplash - which hurts like hell. In fact an
octopus could be used as a
flogger, as I sure some kinky
Japanese does regularly.
Notice that at this point you will be covered in invisible but smelly droplets of octopus juice. Have a quick dip in the sea, to rinse off the smell and chill a bit.
Issues
Throwing is hard: can I hold it by its tentacles and just whack it against to rock ? Just try: it will slip out of your finger, and possibly hurt you in the process.
Someone called me a monster because I was doing what you suggested ! Invite them over for soup. Everything will be pardoned.
Have you ever actually done this ? Many times. Underneath my suave CS nerd appearance, a mollusk-killer lurks.
After the tenderizing is done, you might as well remove the beak - easily done with a screwdriver.