When trying to divorce someone living in a foreign country, service is always an issue. New York requires personal service of the defendant before any other type of service is authorized. The easiest way to handle the issue is to have that person acknowledge service. The acknowledgement does need to be signed in front of a US notary.
The second way is by having them served. Depending on the country in question, the difficultly of this may vary. If the country in question is a signatory to the Hague Service Convention (an international treaty regarding service of process in civil legal matters), you can follow the rules of that treaty. While New York State requires personal service because the Hague Service Convention is an international treaty, it supersedes state law. If the nation in question is not a member of the Convention, you will need to serve that person in the same way as you would someone who lives in a different state, which means that you would need to serve them personally.