Microsoft DNS Across a site to site VPN

In order for Active Directory to work and your sites to see each other you need to have your DNS servers at each location aware of the other locations. You CANNOT use your ISP or provider DNS servers.

That usually means the same HH site DNS server records should be configured in the DHCP scopes on each of the subnets.

This becomes tricky when you are not able to access the vpn during an internet outage as you will lose all internet access unless you have secondary and tertiary DNS that is pointed at a local DNS server.

This the best option is to have an A/D server at each location and use active directory sites and services to manage this for you. Setup your local A/D server as DNS server #1 and setup the server at the HQ location as DNS server #2.

With this DNS setup you have redundancy AND you offload your
authentication traffic to the local servers.

If this dual dns server approach is not feasible then
assigning your HQ site DNS as primary and secondary will have to suffice.

Be advised that while WINS (Windows Internet Name Service resolution) can be made to work across a VPN using H or P node NBT resolver types in DHCP options config this is not a suggested approach. Use AD and MS DNS.

Also your remote sites will need to use FQDN to access remote network
resource.