Personally I'd prefer Localbitcoins over paxful any day. It's more established and more trusted, and there are more advertisements and better rates in general.

However, in terms of privacy I think that paxful may have the upper edge. Localbitcoins has recently implemented a verification system that forces you to submit ID once you reach a certain threshold of trading volume, on top of the fact that they already force you to verify if you want to place any ads. From experience, there are also more ads on paxful that allow you to buy/sell with certain payment methods without needing to be verified.

I don't think that you can go wrong with either, I've heard some complaints about Paxful's support speed but I've never experienced it first hand. The most important thing that you should look for is probably rate if you're comparing the same payment method between the two.