Public network mode sets your Syncword to 0x34. This is the standard sync word for public lorawan networks. If you are not building a private network with custom sync word, you should set this on by default.
If you say private network like deploying your own LoRaWAN Network but uses same sync word used on public network, then there is no problem to that. Although using different sync word adds layer on what device can go to your network, it is not required to be different. To add to that, if you use same sync word as public network, you will have no issue roaming to public network if you decide to roam or to use packetbroker of TTI.