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Yes. DigitalOcean’s DNS service can be used with private networking. The one caveat would be that the Droplets can not be completely isolated with its public interface disabled. They will still require outbound access to port 53 for DNS lookups. In general, public DNS does not care if the IP address it points to is accessible or not. So pointing internal.example.com to an private IP address with an A record will work, but it will not be accessible if you are not on that private network.
If you require complete isolation from the public network, check out this tutorial for information setting up a private DNS server:
I achieved that by following https://gist.github.com/so0k/cdd24d0a4ad92014a1bc, but it only works if the DNS is a real one (not some internal fake DNS).
If the DNS doesn’t exist, I achieved that (accessing other droplets through the internal DNS) letting only DO internal nameservers IPs (198.41.222.173, etc…) in resolv.conf (removed 8.8.8.8 and 8.8.4.4 from it).
The problem is that it won’t work when I try to lookup external sites.
In the end, I still don’t know how can I use Digital Ocean DNS internally with a fake internal DNS, in such a way that it resolves correctly when I ping a host in my internal domain, but uses 8.8.8.8 or 8.8.4.4 for external ones.
It will be great if DO provided some tutorial about how to do that.