No, we are not "immune" from a bite or sting. All of us have skin that can be pierced. The severity of the body's reaction varies from person to person. For example, my mosquito bites don't get as swollen as my daughter's, yet neither of us has an allergic reaction.
An allergic reaction occurs when the immune system overreacts to an allergen. In stinging insect allergy, the allergen is the venom released during a sting.
You may not get an allergic reaction, but that's different from being "immune" to stings and bites. Also, some people are more attractive to insects than others. I like hiking with my husband, because mosquitos consider him a tasty treat and he lures the majority away from me.
