Since there's no way of connecting "private" to a number (unless you can think of a formula that can do it), no. It has nothing to do with the phone or the app - "private" and 123-456-7890 can't be matched by anything I can think of.
And if you could somehow put "private" into the phone number of a contact, you couldn't call that contact, because "private" as a phone number is 7748283 (unless the contact your talking about happens to have that number). But you'd get that ringtone for anyone calling you with a blocked number, which isn't what you want.
(A private number doesn't mean that the number is sent to your phone, but the phone is told not to display it. It means that the number isn't sent to your phone, so your phone has no number aginst which to match the ringtone.)