Sounds like you and your little girl are very compassionate.

And from your question, I would venture to guess that the neighbor's girl has a difficult situation that may be affecting her self-esteem and her social skills.

Your first priority is to your family and your little girl. Explain to your child that sometimes people who are unhappy make choices that are not very good. Tell her that you know she cares for her friend. Then explain that it is also your job as a mother to protect your child.  Tell her that you will try to help her continue to play with her friend but that you are the mom and if the friend continues to take things or otherwise refuse to follow the rules at your house, then you will have to say no.

Make sure that when your daughter plays with the other child that you are there to supervise, especially the next few times.

Explain the rules of your household (speak nicely to others, do not take things that aren't yours, etc.) and say that if anyone violates these rules then the playdate is over for that day.  This way you are not necessarily singling out the other child and saying she is "bad"--you are just clarifying your rules.

You may eventually have to say "no" to this relationship if you feel it is not in your daughter's best interest for the children to continue playing together.

Finally, it is definitely not your responsibility to "fix" things for this other little girl.  However, if you have the time and energy, you have an opportunity to be a positive adult influence.  If you want to, look for opportunities when she might want to spend time with you. Check-out the resource below about ways to communicate with children.  Avoid accusations and try asking questions about her say.  She may open up.  Take care not to overstep your bounds by offering a lot of advice or being critical but there is no harm in listening to the child.

I would not be surprised if she gravitates to your child and household because she craves time with a loving family with clear and consistent structure.

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Candace Lindemann, Ed.M. - JustAsk Expert
Educational Consultant and Writer
http://CandaceLindemann.com

If she only sees her mother very little, then her training has gone lacking. This is an opportunity for you to show her how to respect others property. Since she is six, you can try setting up a play area out side that is safe. Also, you can set up a room or play room and just let the girls play in that one room. You may try reading books about respect to the little girl. But laying down clear rules in your house is the main thing here. You want to stress to the child that she must follow your rules or she has to play outside (in that area they can play in). And just be consistent about it. Your child should be told that the little girl is learning respect and she has to help her out by following the rules. You can even make a chart stating specific rules eg. "No running in the house", "No going into (your daughter's room) without permission", "No eating candy on the couch"....