Hi Suzanne:
Yes, the first time we used private babysitting was aboard a Celebrity ship when my daughter was almost three years old. We checked in once to make sure everything was okay (which it was...) Celebrity, as with most other lines, uses room and bar stewardesses and also front desk help as babysitters. We have used private babysitters on two other cruise lines and have found them all to be very sweet young ladies who are happy to be around children. (One aboard a Norwegian Cruise Line ship even left a present for my daughter to open when she awoke in the morning...now that's service with a smile!)
I checked with Celebrity and they don't have beepers to give to parents of small children. (Some lines do such as Disney Cruise Line and Princess Cruises.)
I hope this makes you feel more comfortable with leaving your daughter with a babysitter. We have found the private babysitting fits more with Alex's early to rise and early to bed habits compared to group babysitting cruise lines offer for those children old enough to be in the youth program (which is three years old in Celebrity's case...)
Bon Voyage!