For me, personally, I need to 'know' the parent/Mom first. Making sure she is not a nut or a weirdo or simply a noxious personality. Before I let my kids, play with her kids.
Then, even before my daughter was that age and from when she was in Kindergarten, I ALWAYS explained to her... about how to 'discern' and choose friends. She is now a very wise girl, for her age. She is in 4th grade now.
Once I do feel comfortable about a Mom/parent... (because I go by my instincts about people), then sure, my kids have play-dates at our house or theirs.
IF your child goes to another person's home... you need to trust that/the parent/the household and be sure of that. OR you stay. I always, let the other Moms stay if they want. Or they can drop off.
AND yes, when other kids/Moms come over, I SHOW them around the house. I EXPLAIN my supervision to them etc. and the rules. And that, I will.... correct any kids misbehavior. But the other Moms/kids I know, they KNOW me and are fine with that.
And yes, YOU as the parent, ASK the other Mom, what the kids will be doing, WILL she be around (some Moms just don't supervise at all and are not around when kids are over). So simply, ASK. That is your kid after all.
And the basic common sense is: IF your child is a friend of another child... you NEED to know the other parent/Mom, too.