I have been doing playdates weekly with my son since he was about a year old. I met a wonderful group of moms at a Mothers' Group after our children were born and we remained friends. We would alternate the location and meet once or twice a week. We would meet for 2 hours or so. It was always acceptable to come late or to leave early. We always brought our own drinks and snacks, but of course had enough extra snack to share with the other kids. My son is now 4 and I have a 2 year old daughter. I totally trust the people who we are with and I would leave my child with them if I needed a sitter, but not for a playdate. The other moms have their own children to watch and I don't think a playdate is the time for them to have the added responsibility of keeping an eye on my kids.
Have fun! Playdates have been wonderful for my kids, but also for my well-being at a stay-at-home mom. The chance you get to connect with other moms who are sharing a similar experience forms a lifelong bond!