For a PBN? Private would keep Google from knowing the same person as xyz.com owns the domain. Allegedly Google MAY be able to see though private anyway... people have floated this idea, but its not 100% certain. For safety I'd go private for PBNs if the same name on the money site registration. Some even (many here) use alternate names on PBNs to avoid buying private.
In general private will also garner less spam to the email you use. People scrape registrations and spam then sometimes. I prefer private for convenience.
No "SEO" boost though.