Giving a child what they want when they are misbehaving just reinforces the bad behavior.
You mother must insist on being respected. When your little sister, is disrespectful in front of friends with your mother it is important for your mom to embarress the heck out of her. Her little jaw needs to hit the ground by whatever action your mother takes make the peer pressure work for you. If she wants to act up in front of her friends she should be prepared to suffer the consequences. There are so many tools parents have in their arsenal to get children to behave. Is there some baby picture she wouldn't like her friends to see "the money shot" (the one picture the kid would pay money for friends not to see). Whip it out. Or trying just calmly saying no. No matter what she does just say no and later when she wants to spend time with her friends you tell she can't and it is because she did or said ABC or 123.
Parenting isn't for the feint at heart but for the brave. If your mother isn't up for the task, you may be better equipped to handle this depending on how interactive you are with your sister. If you were willing to spend more time with her and her friends, you could also be a useful tool in correcting incorrect behavior. If she does something inappropriate with your mother, you discipline her but she knows the discipline will come however it will be most effective and appropriate coming from your mother.