If multiple people live on these private named roads, I leave them on there for the reasons Alan noted. There are a few out in the country around here, and calling them "Pvt Rd 352", for example, allows the TTS to read it properly, so I've been doing that.
In some cases, I've found that properties on that road are actually addressed with the private road name as the street address, i.e., "3152 Private Road 12". If it's an address someone could navigate to, it definitely should be shown on the map.
AM for Baldwin
-Mobile Cos., AL // Grenada-LeFlore-Carroll Cos., MS // Escambia-Santa Rosa Cos., FL // CM for the USAHTC Droid DNA (Android 4.1.1) on Verizon Wireless