If the person spewing the hurtful insults is a loved one or close friend, I deal with it by walking away and avoiding them until they apologize. If they happen to live under the same roof as me, I leave without letting them know where I'm going and I go stay somewhere else until I get a text or call from them that's an apology.

If said person is an acquaintance or stranger, I let them know how I feel about what they said in a direct to the point manner without sugar coating anything. If they don't like what I have to say, I take measures to remove them from my life such as blocking them on social media and blocking their phone number (if they have my number).