If a private, third party wants to fully fund one's PhD studies, how would one mention this in an admissions essay?
Typically you will be asked about your funding sources at some point during the application process. That would be the right time to click on the box saying scholarship/sponsorship/private funding/whatever. This is true if the application to the university program is separated from that of scholarships/funding opportunities. On the other hand some institutions don't allow their students to pursue a (graduate) degree without having secured a scholarship. Therefore in that case you'd be applying to the program and to the scholarship at the same time and with the same dossier.
Having said this it definitely will not hurt to mention your funding sources in your personal statement if you think that this would add value to your application. If the third party is a well known charity or foundation whose scholarships are indeed prestigious awards then you should probably mention them. This however wouldn't hold if your sponsor is, say, your grandfather.