There are these options, as far as I know:
Open a private chat room for this purpose, give this user the right to join, and super-ping him there.
This is the preferred way for minor things.
Send a moderator message, with the moderator tools. It sends a e-mail to the user, and a copy to your fellow moderators (and yourself) and the SE team. It gives you the option to also suspend the user for some time, but it also works without giving a suspension.
Use this for major infractions.
Send a email. If you do this, also send a copy to the community team, so they have an overview about what is going on.
This should only be used if the other ways are somehow not effective.
Of course, often you don't really contact the user privately - a public comment on a post or a ping in your public chat room often work, too. (I think we never had to contact anyone privately on Crypto SE.)