we can leave aside the homegroup (simple filesharing through a common password) and the domain (requires a domain controller) in most cases, so private and public ist left.
as posted above both network types just enable a different set of firewall rules - you can see the details in *Windows firewall with advanced settings*.
more important than the network type is to not ignore or cancel the dialog which pops up after the first start of VE PRO asking for firewall exeptions beeing added.
if your network is set to public it is highly probable you're on a machine which has more than one NIC (Network Interface Card), also called a multi-homed computer, so any performance problems might be the result of wrong or conflicting entries for gateway in your network settings.
note. using multiple networks (eg. one for internet and one for audio) requires correct cabeling and setting of IP adresses and gateways. do not forget that WiFi/WLAN is also a network ... networks without gateway are always public by definition, but you should never set additional gateways without understanding network communication.
one of the system requirements for VE PRO is a *working gigabit network* and since network conigurations are legion you should at first get a clear picture how your devices (including modems, routers, switches and printers) are connected and should communicate.
if you are asking vsl support for troubleshooting your specific network problems it is absolutely neccessary to provide a precise description (even better a drawing of devices, cabeling and IP adresses) to get a clear picture where a problem could be even located ... *does not work* or *has bad performance* is definitely not enough.