A tough situation - I know because I have been the mom of the "bad" child. My boys who are only two years apart went through a "bad" phase - for about a year. That being said, they weren't bad or naughty. They were very high strung though and I was constantly trying to figure out how to calm them down and discipline them - especially hard at other people's homes or in public. About 9 months into it I had finally unlocked the secret - they were responding to discipline and we were mostly only having the typical toddler tantrums like when they want candy at the checkout line - but even those, man I've seen much worse than my children.
My best friend also had two boys, both about a year younger than my boys. We always loved getting together so the boys could play and we could sit and chat. The playdates weren't without incident of course - they're boys! There were of course bruises, crying, arguing over toys, etc.
Best friend and I attended a birthday party for a mutual friend. I was w/o my husband at the party (2 supervising eyes less), with boys in tow. My older son had been extremely tired that morning from a field trip the day prior and I contemplated skipping the party because I was afraid that he might cause a scene - but the mention of going to a birthday party put him in such an elated mood, and we discussed the rules a few times prior - so I decided we'd go. Ugh - there were a few incidents - but that is all I'd call them. Older son threw a fit because his hot dog fell out of the bun - I took care of it. Younger son stuck his finger in the birthday cake and when best friends husband pulled him away from the counter younger son briefly hollared at him. And the third that I am aware of was my boys dumping sand into the motorized kids jeep - a situation which I stopped immediately upon seeing it occur and checked to see if anything was damaged - no. Of course, when it was time to go they both got upset and I had to drag them away. As embarrassed as I was about their behavior, I had only myself to blame - we should have skipped the party. And I hoped that the adults and host would understand that we had a rough night.
The following day I got a phone call from best friend. She in so many words told me that her boys would no longer be able to play with mine because she was afraid that they would influence hers negatively. Up to this point she had already quietly been excluding my older child from hers because he was the more roudy of the two - an action which I was willing to let go because it was convenient. She told me that the host had called my boys "hellions" and that there have been many times she has been embarrassed and ashamed to be around my boys because of ways they had acted. I admitted that the past year had been tough - but I was seeing marked improvement and felt hers and definitely the host's assessment was extremely short sighted. I was angry. She failed to see the improvement I had made with my boys. She was holding grudges against them (you don't do that to kids!). And being that we had only lived in this area for about a year - she was removing my children's best friends from their lives. She was giving my children (and by extension - myself) an ultimatum to either shape up or ship out. (Do best friends do that?) She said we could still hang out toghether, w/o the boys. Really? Does that make any sense? Two SAHM moms who's husbands both travel for business are really going to pay for babsitters so we can sit and chat? Not on my wallet - backyards are free.
DRAMA for the next few weeks. I ignored her for a while. I refused to let her hear my anger and just told her I was sad for my kids. I discussed the situation with my other friends who told me she and the host were way out of line. I decided that we were no longer friends.
On some of those phone calls she admitted that she was holding pent up anger/resentment/fear toward my children and me. She was afraid that they would hurt her boys. She didn't think that we disciplined them and that they were allowed to lie to us. (The latter sentence somewhat true in the beginning of the tough phase - but not at all relevant at the time of her and host's assessment.) She had also heard my son say something to hers - a not so nice phrase he had learned at Kindergarten - that she chose to take out of context and let freak her out. NONE of which she had ever confronted me about until the blow-up.
Best Friend's husband arranged for both of us to sit down and talk things out face to face, with husbands at our sides. My husband was on my side. Hers was a mediator - her husband has a wonderful gift that way - he's extremely logical and true-serving.
Tears were shed and we worked things out. But I no longer consider her my best friend. I do not hold her in such high esteem as before. Playdates rarely occur. She hasn't because she obviously still holds something against my children, but if she invites us over to a group event with other families I will politely decline. No need to cause her to stress that my boys might cause a scene.
I only relate the whole story because I think the order of it is important. I really feel that she and I would still be best friends had she just approached me about her concerns when the incidents occurred. Had she been open and honest w/o passing judgement or placing ultimatums - then I could have taken her concerns and used them proactively to improve my children's behavior and my discipline techniques at the time the issues occurred. Instead, she let things bottle up - incidents and issues that I was never made aware of - and then she used them as weaponry against me.
I've forgiven her for her mis-sight, but I am always on pins and needles now. If my children in any way bruise or hurt hers...or if they talk-back to an adult...or if who knows what happens and she doesn't approve...is the friendship going to take another turn for the worst? So we limit our physical appearances. And I keep our conversations petty and light. I don't discuss many flaws or parentng dilemmas in depth because I'm afraid she is putting them away in her vault for the next time she decides it's time to "intervene."
So my advice would be this:
Bring it up to your friend. Let her know that you've seen her daughter kick yours a few times - intentionally. Be honest. Don't say that you are going to limit playdates because you know you don't want to do that. She either knows about the behavior or doesn't. Either way she will pay more attention now, simply because you brouht it up. Say something like, "I know we don't always see what our kids do - we can't watch them all the time! When we see eachother's daughters mis-behave, how would you like to handle it? Do you want me to tell you right away or do you mind if I step in and put yours in time-out, and vice/versa." Or whatever mode of discipline you both agree on.
I think there are too many people that think that parenting and discipline are only the responsibilites of the parent of that child. When in reality, children need to hear from all people/adults they've offended what it is they did wrong and the consequences, just a they need praise for good behavior. So they can not only learn to respect others, but be respectful of the rules of others.
Post edit: I just noticed that you also posted about a terror of a child. Is this the same child who kicks? If so then you've got another situation on your hands. It's not just about poor behavior now - it's about your child's safety. The mother of that child is obviously not being a mother. The child is bored and has no boundaries. If your child is constantly in harms way then you have to make the decision whether or not to hang out with them and I don't think there is a decision to be made. I still refer back to my previous suggestions though, with the exception that it would be right for you to keep your daughter at a distance. Just don't give the mother an ultimatum "that she has to earn the right to hang out with you / your children". Sounds pretty silly doesn't it? Just be honest - always the best policy.
"It takes a community to raise a child."