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This thread is resolved. Here is a description of the problem and solution.
Problem: ATE - where does the data live
Solution: The WPML Advance Translation Editor translation itself is being done in a special server(external) that is controlled by WPML Team. When the translation is complete the data is being saved in the site's DB. We will be using WPML Advance Translation Editor for just translation and the translated data will be stored in the client site. So we don't need external service to show the translations.
Relevant Documentation: https://wpml.org/documentation/translating-your-contents/advanced-translation-editor/