The 10.10.10.10 server needs routing table entries to figure out how to contact the 192.168.1.0 network. If you're also using the 192.168.1.0 gateway for internet access, you need to tell the 10.10.10.10 server to use that gateway, too.
The DNS entries should come from the DNS server you're using, not the HOSTS file. That way you only have to update the entry in one place; otherwise, you will have to edit each client that tries to connect to the 10.10.10.10 server.