Here is some info to help you figure it out:
https://www.irs.gov/businesses/small-businesses-self-employed/statutory-nonemployees
"Companion sitters. Companion sitters are individuals who furnish personal attendance, companionship, or household care services to children or to individuals who are elderly or disabled. A person engaged in the trade or business of putting the sitters in touch with individuals who wish to employ them (that is, a companion sitting placement service) won't be treated as the employer of the sitters if that person doesn't receive or pay the salary or wages of the sitters and is compensated by the sitters or the persons who employ them on a fee basis. Companion sitters who aren't employees of a companion sitting placement service are generally treated as self-employed for all federal tax purposes. However, the companion sitter may be an employee of the individual for whom the sitting services are performed; see Pub. 926."
Since the agency does receive the pay from the client and pays the sitter, the agency appears to be the sitter's employer (W-2, not 1099).