I've been reserving 'no' for the important things, like bringing something heavy down onto his head, potentially or actually (one day he'll see cause and effect!), from a high surface! Redirecting has been my tactic, as has the rephrasing into 'leave it', 'put it down', 'let go' (in Spanish and English) and 'gentle' (in Spanish and English!) - the last two usually used for when DS grabs the little dog by the hair and reefs it...
I'm sure it would somehow make other people feel better just to see me say no and maybe give a smack on the hand, but that won't actually teach Oscar anything more than to hit when he wants something from someone else! When I take something he shouldn't have out of his hand, I ask him and say please, so that he doesn't learn about snatching from me.
The other thing I try very hard not to do is to teach about personal possessions. I don't want him to go round saying 'mine' and getting upset when someone takes something from him more than he needs to when he reaches that developmental stage. I know we don't even live in that sort of society, though it would be really lovely if I could instill some Aboriginal philosophy of objects into him - which is, that you don't actually ever 'own' anything (from my understanding, at least).
I see a lot of friends with older kids get really upset when Oscar takes something of theirs to play with that they were not even remotely interested in, until Oscar started to play with it. I realise that kids will go through this, but I wonder how much of it is imparted by telling kids that things are 'theirs' and placating them by saying 'this one is yours, and not x's'. That's why I like the saying "Not for Oscar", as you have suggested, instead of 'not Oscar's', if that makes sense!
I'm rambling, sorry!