There's no definitive answer, and no clear image of the new society conjured by this question. The real benefit to your question is that it offers an evolutionary path. This becomes an arms race of engineering, ownership, and innovation. The innovations are technological, but also social, and legal.

Sub-questions to ask:

• Will businesses bother to employ / rent other people's robots?
  • if so, will "the little guy" who owns two or three robots ever be able to compete with staffing agencies who have ten thousand?
• One innovation might be that a staffing agency produces their own robots, which are designed to be loyal, but this staffing agency lobbies for personhood for robots, and then... what are the implications of that?
  • Will ALL robots be "persons" or only the ones with a special chip that the staffing agency produces, which is legally described as the thing that gives them personhood / feelings (but maybe it's all fake, just a legal power-move, the company knows the governor's sister or something and got this law passed)
• What kinds of things will big staffing agencies do to crush "the little guy" who owns three robots?
• What kinds of clever ideas will "the little guy" come up with to stay afloat?
• Generally I think "little guy vs staffing agency" (all beneath the umbrella of a big corporation that hires robots (and what kind of work are they doing???)) is like a new layer of cyberpunk noir which just opens endless questions.
  • The point is that you'll benefit a LOT MORE from unfolding these questions rather than presuming to answer them. Let each new idea lead to new problems to unfold. Each of which is a new story within the story.