Just to add something...I don't think it's up to the OP whether or not her friend can contact her facebook friends. OP's friends have to decide to accept MLMgirl's friend request. They can say no. If they say yes (they either already know her, or will friend anyone) and it's up to them to deal with the consequences of saying yes. They can defriend her just as easily.
The only problem is if the MLM friend adds people to a group without permission, which several MLMers of my acquaintance have done. I'll wake up in the morning, have fifty emails, and find out they're all coming from a facebook group to which I was added without my knowledge or permission.
But if she's not doing that, just going through FB's friending or messaging function, it's annoying but not necessarily wrong.