No, it doesn't. There only communication is happening through the site. The site issues a challenge to the PC, the previously registered phone confirms to the site that the challenge is met, and from now on the site trusts the PC. The PC and the phone can be on different continents.
The problem though is that you have to do this for every single site you access. So if you have 100 log ins and are switching PC or phone, you'll have to do this same dance 100 times in the next period. And of course, if you're switching because you lost your one device that was registered this way...
