Do the caretakers feel that having a person in the house to do things is an invasion of their private space?

Never bothered me to have help in the house with or without the caregiving responsibility. Did my share of my own housekeeping and not ashamed that I no longer have to do that! When Mother was with us, she had a companion once a week so I could keep my own doctors' appointments and continue my volunteer work.

My mother called dads companion the "baby sitter"...

I think that after awhile everyone gets comfortable with the situation... It is what it is...

I think it depends on the person and the caretaker. It's not an easy answer. If I were home and had someone coming in, I'd feel my space was being invaded, but that is me. Some people enjoy someone coming in, it gives them a chance to socialize and they get help with a bath or household chores.