Internally your OS resolver will be configured to point to a different DNS than anyone external to your network. You essentially have a private branch of the DNS at that point. The only real potential for conflict is when the FQDN of an internal server is the same as for an external server. In that case your internal DNS will serve up a record pointing to your internal server, and external users will get a record pointing to an external server. If staff want to order fresh burritos over the Burrito-over-TCP protocol that will have to use separate equipment. But nobody outside your intranet will ever see a name conflict.