I'm a retired police officer and I'll tell you it's extremely important for you to know the other child and her environment before allowing your child to be in their home.
The way I handled this when my daughter was in grade school and the way she handles it now is to first get to know the mother well enough to talk with her about this issue and then to visit with your child in the home the first time.
Among the things you can learn just in general "getting to know you" conversation is who lives in the home and who they are: ages, activities, occupation, relationship to child, etc. Then talk about what sort of behavior you each expect from your children and from the other mother while your child is with her. How does that family discipline. What are the rules? Who supervises and how closely? What sort of play will they do? What sort of discipline does that family use? Meet the mother and/or anyone else who is around and see the home situation. Trust your judgement.
One of the probably most difficult questions is to ask if there are guns or any other dangerous objects or situations available to the kids. If your child is to ride in their car, ask if they have insurance, if they have a child restraint seat and about their driving history. I never let my daughter ride in someone else's car, even as a teenager, unless I felt sure that she would be safe. I'd start the conversation by asking indirect questions, such as what they think about the law that requires insurance or more directly by saying that I was concerned about traffic and safety and perhaps telling about an accident in which the "other" driver was at fault.
As you can probably understand, this getting to know the other family takes time. I don't learn this all in one conversation. Your child's physical and emotional safety is worth it.
If the mother is also conscientious she'll want to know these same things about you. If she is hesitant to answer or shows discomfort with the questions or your "gut" tells you something may be wrong, don't allow play dates at the other child's house. Suggest that the other child could come to your house for a playdate first and invite the mother along.
My daughter was careful enough that she wouldn't let her 7 yo daughter go to the YMCA daycare director's house until she got to know her better. The first playdate was at my daughter's house. The YMCA director has had police background checks and extensive training but yet my daughter wanted to be sure. The director told me that she was comfortable with the way my daughter handled it. And the director spent time talking with both me and my daughter before letting her daughter go to my daughter's house.
Another way to learn more about a family is to talk with other mothers. You can start a conversation just by saying your daughter wants to have a play date with "so and so." If that mother knows the other mother she'll probably continue the conversation.
It's entirely legitimate, while they're alone, to ask the teacher or even stop in the office and ask them what they think. They cannot directly give you any negative info but they might quietly say that's not a good idea. When I've done this I've started by saying I know you can't say anything but I'm wondering about arranging a play date for my child with "so and so." They might not say anything but they'll respect you for asking.
When I was helping my daughter find day care for her daughter to use when I couldn't I stopped in the apartment office and asked them about the woman who had volunteered to do that. They just shook their head. I did sometimes stop in to just chat and so in a little while they told me that frequently people have filthy apartments and don't supervise their own children in a way that let me know that this included this woman without exactly telling me it was this woman. I know that having this sort of conversation can be tricky and I'd like to think I've perfected it over the years but I do remember doing it as a teacher and then a police woman and feeling
uncomfortable about it. I think that most people do look out for children's welfare and will frequently find a way of telling you without telling you if they're not comfortable just coming out and saying it. It's a matter of you providing the opportunity. The main concern for professional people is the possibility that you will gossip and/or use info for a lawsuit.
Please know that you are just being a careful parent and that you need have no concern for hurting anyone else's feeling's or for them to take offense. Your child's welfare and safey is of the highest priority.
I would limit the amount of time that your child, at 4, would spend without your supervision. My granddaughter usually wasn't comfortable to be alone with anyone else's family until she was 6 or so. Therefore all of her first playdates and most of her later ones were with both mothers present until the families knew each other. It was as if she had an inner need to know and see that her mother trusted the other child's family and home environment before she felt safe. This wasn't anything spoken. It's just the way it happened.
I suggest that you and your daughter meet the other mother and child at McDonald's and share food and company. That is enough for a start. If you feel comfortable with that situation take your daughter to the other child's house and stay awhile. If you still feel comfortable you can leave for 30 minutes to an hour and go back and pick her up. Be sure to exchange phone numbers. And be sure to tell the other mother what you expect in the way of supervision. For example if their is a teenager in the house is it OK for the mother to leave them alone with the teen?
After thinking and writing all of this my recommendation is that at age 4 all play dates include both mother's for an extended period of time. And when you do decide to leave your child in the other mother's care be sure that you both agree on the boundaries for activities, discipline, and amount of supervison. Make it clear that if your daughter at any time wants to come home that the mother should call you and you will pick her up.
Always pay attention to what your daughter says about her friend, the family, what they did, and how she liked it. If at any time she says she's uncomfortable for any reason do not have playdates without your presence. It may be that she's just feeling insecure and that nothing is wrong but you cannot be sure how that family reacts to her feelings.
Hope I didn't completely scare you. I do have strong feelings about children's physical and emotional safety. But I've also gave my suggestions more detail than I sometimes use myself. I have learned thru experience that I can trust my own intuition and shorten the process at times.
I would be glad to talk more with you about this if you'd like.