Google's John Mueller said that having a private whois domain name registration status does not change how Google trusts the site. It really does not impact the site's rankings in Google Search.
He was asked on Twitter "from SEO pov, Does keeping WHOIS information public help in website Trust and Authority?" John responded "it doesn't change anything." John latter added that "making a domain registration private doesn't make the domain a part of a link network."
Here are those tweets:
It doesn't change anything.
— 🍌 John 🍌 (@JohnMu) November 24, 2020
Making a domain registration private doesn't make the domain a part of a link network.
— 🍌 John 🍌 (@JohnMu) November 24, 2020
Google is a domain name registrar, has been since 2005 and back then, SEOs asked if this would matter. But no, it really does not matter if you have public or private whois data on your domain name.
Forum discussion at Twitter.