I have not felt comfortable asking someone I have known for a short time to watch my children, so I haven't.
I do, however, have my children spend the night with one of my very close girlfriends at least once a month. I take her two children for regular spend the nights as well. It's a fantastic trade! I get to enjoy my space (and my husband) when my kids are at her home, and also get to revel in kid-dome when her kids are with me!
Also, I love that my kids can spend time with people who they really care about, and who care about them. My kids have an extended family of sorts with our close friends. If I did not create it, they would not get to experience extended family. And, I really like the, "it takes a village", attitude.
I have a new friend who just moved to our town to be with her partner, who is an old friend of my husband. While I don't feel comfortable having my children at her home, I am more than comfortable having her daughter at my house, and have had her over for spend the nights many times in the last few months. I like being able to give other parents a break, and love watching a troop of toddlers run around on the beach. My kids LOVE spend the nights.
My mom also lives in town, and my kids go over to her house for a spend the night every few weeks. It's not so that I can do anything particularly special/an emergency, but so that I can recharge and have space, which makes me a better mom. I love spend the nights (as does everyone involved).
ETA: I thought I'd share my experience, since it seems a bit different than some other folks :)