Heroes: do they share weapons?

I havent played NF yet and I have question - I was looking for answer but could not find it

As i read every character on my account will have his/her own "version" of Koss, Dunkoro, Melonni and so on.. Every hero is different BUT do heroes with the same name (but on different player characters) share the same weapons?

Here is an example to make my question a little clearer: if I play my monk and give some sword to my monk's Koss hero, then relog to play my necro char - will necro's version of Koss have still the same sword? Or maybe I need another sword for my necro's Koss?

heros are character based.
If you have Koss on your monk, your koss on your necro will be a seperate character/thingy

So no they do not, you would have to storage it and then equip it for the other one.

hmm with 8 PVE chars.... thats 16 warrior heroes. Thats 16 Superiorabsorbtion, 16 swords, 16 vigor runes.... wow

First, thank you very much for answer.
Second... omg... so I will need now every weapon I get, lol

Ye, lol.
BUT 16 warrior heroes also mean 32 additional storage space for weapons/focuses/shields

i think heroes r char-based, not account-based

i'm not sure how i'll be able to farm for shields/swords/staves for all my new heroes now after AI update, but i guess its still possible... and there r collectors, i dont like collectors staves - 1st i have to farm collectibles for the staff, then i have to find upgrade parts, it takes too long, i'd rather farm some green staff and be done with it but warrion collectors items r fine i think

Not really, you share the same inventory.

That's true; heroes do share inventory. But weapons equipped on heros do not take inventory space, so you can in fact use them as weapon storage, if you can put up with the time-consuming micromanagement. Personally, all my caster heros are getting Insightful Spiritbinders until I aquire something else for them. Yay for /bonusitems.

Does anybody know if the heroes get penalty for using customized weapons for my character? Also, I got to the part where I get to choose a hero to train (using my Tyrian character). I can choose an elementalist or ranger, but will I be able to get the other hero later in the game? I don't want to make the wrong choice if I can't.

I believe its mentioned that each character will NOT be able to get all heroes. Like that choice you will have to decide and can always get the other with a different character.

Also i dont recall seeing/reading about penalties for using customized weapons =/

Once you complete the game you can go back and get the Heroes you missed out on. That was confirmed by Gaile.