Private preschools/daycares DO have to have background checks. IF they are licensed. I have worked in a private christian preschool for 11 years and I know that this is the law.
As far as CPR/first aid - there has to be ONE certified adult in the room. The school I work for (one of the largest in Dallas county) we have 2 - each and every teacher must be certified - standards set by our school board.
You do not need special training to begin - but you do by law have to have an orientation. After that you must have 15 training hours per year - that year ends exactly one year from your hire date.
Legally you can and will be held responsible. You can be held legally responsible if you so much as put a diaper on a child that was not provided by the parent.. and the child has an allergic reaction to it. I am not making this up - it has happened. Not in our school, but we have had training classes on this stuff. The parents sent one diaper for thier toddler. This child then had a "sick" tummy and went through that diaper very quickly. Attempts were made to contact the parents - but they were not answering phones. So the teacher did the only thing she could do - she put on a school provided diaper. Turns out the child was allergic to something in the diaper and had a horrible reaction. Spent time in the hospital. The parents sued and won. Makes absolutely no sense I know - but now that poor teacher was charged with child neglect and endangerment and is never allowed to work around or with children again.
I can personally tell you - if you get into the early childhood teaching career again GO BY THE BOOK. Most parents are amazingly wonderful. It can be the most rewarding thing you will ever do. But - speaking from personal experience it only takes ONE lunatic to cause you a lot of heartache... even if you do go by the book (some parents just like to lie).
Anyway - this is not a rant thread for me. Just follow your heart and do what YOU feel is right - for the children - not for the director/owners of that school - for the precious kiddos.