No.

The advertiser pays the whole fee.  The customer pays only the listed price.

If you create an ad to buy bitcoins then you will pay the 1% fee, and the seller that responds to your ad will receive your advertised payment.
If you create an ad to sell bitcoins then you will pay the 1% fee, and the buyer that responds to your ad will pay your advertised price.

Because the advertisers are generally aware of this fee, then typically adjust their advertisement to compensate themselves for the cost of the fee.  As such, it is actually the customer that is indirectly paying the fee.  An advertiser that wants to receive BitStamp+5% for the bitcoins that they are selling will typically sell on localbitcoins for BitStamp+6% to cover the costs of the fee.

The fee is only charged if the transaction is completed using localbitcoins transaction service.  If you meet the person face-to-face and the bitcoins are sent directly to the customer's wallet instead of going through the localbitcoins transaction service, then there is no fee charged.