There is no reason to be concerned about Localbitcoins sharing your address. And no, buyer does not see it when you open an advertisement on their platform.

They simply don't do it because it makes no sense for them to do that when there is already a middleman which is themselves who are handling all the transactions. You don't ever send funds directly to someone else's address.

I don't see a reason to be this concerned about the privacy of your funds when dealing on localbitcoins, but if you want, you can certainly use a bitcoin mixer when depositing funds into your localbitcoins deposit. That way, even LBC themselves won't know what your actual address is that sent the funds. But if I were you, I wouldn't worry about this unless the amounts are large.