In general, “Private User” shows up for people who are Guest users.
See if this post (or the resources it links to) help explain your scenario:
In particular I’d also call your attention to this excerpt from the Guide page on Guests:
If two people are seeing each other as a “Private User”, they are both Guests. If Guests see this, that means they are not working in either the same team or project together; this is to ensure that if you’re working with clients, they cannot see one another’s names unless you want them to. Once they are both working in the same project or team, their names will be displayed for one another.
In order for Guests to see one another’s names, they must be Members of at least one project together. Just add both Guests as Members to the same project and they will then be able to see each other’s names.