If one spouse meets the residency requirement of a state or country (such as having lived there from six months to a year), a divorce obtained there is valid, even if the other spouse lives somewhere else. The courts of all states will recognize the divorce.
However, decisions a court makes regarding property division, alimony, custody, and child support may not be valid unless the court had jurisdiction over the nonresident spouse. The court gets jurisdiction when the nonresident spouse is personally served with the divorce documents (meaning they are delivered into the person's hands), or consents to jurisdiction. A nonresident spouse consents to jurisdiction by showing up at a court date or signing an affidavit of service, acknowledging receipt of the filed legal documents. It can also happen if the nonresident spouse abides by the rulings of the court; for example, by paying court-ordered child support.
If you receive documents from a foreign country, you may want to consult an attorney about whether your state court or the foreign court governs the issues. This depends on many factors, such as which particular country is involved, where the parties lived and for how long, and, of course, whether children are involved.
For clear answers that can help make your divorce simpler and reduce your expenses, see Nolo's Essential Guide to Divorce, by Emily Doskow (Nolo).