Do you ever tell this woman "I'm sorry, but that just won't work for me. I'm happy to drop daughter off at your house, or we can arrange another time"?
I think it's poor form for the mother to be 'protecting' her daughter from dealing with the realities of life. She should be teaching her child that toys are just toys, but it's friends that make the toys more fun-- instead of indulging this behavior. How long does she intend to indulge her daughter's behavior? Because what she's doing, in essence, is fostering it--likely because she doesn't want to deal with the tantrums and crying and work of helping children to share.
You should speak up when she says something about sharing: "You know, it's good for her to have practice. Have you thought about putting her favorite toys away before a playdate, so she can get along with her friends while they're over?" If it were me, I would only say 'yes' to hosting when it works for you...
Also, just be cautious of the other mom and what/how she excuses her child. Over the years I've been on this forum, I've seen relationships which probably started a bit like this (with one parent/child pair seeming emotionally responsible for the other pair's needs) and have ended up in situations where bullying was condoned by the other parent-- that is, the parent of the child who behaves in a bratty way or says mean things gets excused by their parent and it becomes expected that the other mother/daughter should follow suit. When friendships become truly unequal--when one party is expected to accommodate and be compliant to the other's needs without reciprocity-- then you are in for a whole mess of further dysfunctional behaviors.