I think Localbitcoins beats most the exchanges in terms of Fees pretty well. All you have to do I get a verified seller with a very good reputation and has done so many trades overtime. Say +3000 or +10,000 trades with different partners.
Apparently LocalBitcoins Charges about a 1% fee for using their service all you have to do is worry about the price of the seller. Now if you compare this to the likes of Coinmama(charges 3.5%), Bitstamp(charges even upto 8%), CCEX(2.9%)... after looking through all that fee i would think LocalBitcoins is the best option.
Secondly, it's a little bit more private compared to other a venues of purchase.
i have used Localbitcoins for a while now but haven't face any problems yet.
There are other Peer to Peer Markets like
LocalEthereum https://localethereum.com (deals with Ether which you can transfer and Buy BTC. Added advantage is that Ether's Transfer fees are usually lower
and then
Remitano https://remitano.com/ but am not sure about the liquidity of the markets depending on the country you are located in.
Verdict i would go for peer to peer exchanges any time any day