My source attributes her folk notion of dog’s licks curing mosquito bites to something she was told as a young girl in Korea. In Korea, during the hot seasons, there are many mosquitos, and therefore mosquito bites were a constant annoyance during this time period. When her mother caught her scratching at her mosquito bites too much, she told her to get the dog to lick it, because that would relieve it. Being young, she followed her mother’s instructions and was convinced of the efficacy of the treatment. When I tried to ask why that would even work, she simply said that she didn’t know, but it simply did. I believe this to be a common trend with folk remedies.
In my opinion, this is a classic case of placebo effect. In her head, this remedy was going to work, and therefore when the dog licked her mosquito bite, she convinced herself it was going to itch less and therefore it did. Whether or not dog saliva contains anti-inflammatory medication, I do not know; however, I find that simply forcing oneself to believe that something is not itchy is something that the brain can do on its own.