I am the cleaning lady !
I am also the cook, babysitter and the maid.
How nice would that be to have someone else besides me come clean my house. I hate cleaning and spring cleaning is the worse.
I dont know how I would feel letting a stranger come into my home whether I am here or not. I think knowing that they are with a legit cleaning company with great references and knowing they have been bonded/had a background check would make me feel much better about letting them roam my home freely. I'm sure that you will pay more but "you get what you pay for" paying for a peace of mind and trust goes a long way.
Maybe you could ask for a resume with references and do some interviews. Seeing someone work history in black and white and making a few call could help you find the right person.