Do I get to keep them? If so how do i find them?

They're in your inventory. You don't have to throw them if you don't want to

they are not in my inventory. Is there a specific inventory? I mean the ones that are thrown at me.

Then I guess you didn't buy any? You have to buy them first. Then you can go to your inventory and you can see they're there

EDIT: I see your updated post...you do not keep the snowballs that are thrown at you. Sorry for the confusion

doing a screenshot would work, if you want to see the effects for your own amusement, but there is no 'saving them" in the game once their thrown their gone. It's not like a pillow fight.

There used to be a thread in Human Avatars that showed them all and their effects, but that was in the old boards and I dunno if anyone has recreated it.