Have an Answer?
An interesting question and one which I am sure members of each branch will have a different answer. Let's apply a little logic to the question. First being an officer in any branch may offer more opportunity to interact with a local populace on more than just a social or commercial basis although there is clearly nothing wrong with dealing with locals in less than an official capacity.
Next, consider who routinely has access to the people of other countries. Both the Navy and Air Force function from bases/ships generally at some distance from other countries while the Army and Marines tend to be based and live in foreign countries for extended periods of time. These are just general rules and you will find some representatives of all branches on any large deployment by either the Army or Marines.
Almost everyone of any branch that I have known have had or taken the opportunity to travel to other countries from their duty station by using open flight seats at no charge on military flights (this information may be out of date). When I was in Asia a friend caught a combination of military and commercial flights to backpack in Kenya and India. He took two weeks leave each time.
Another friend stationed in Germany was able to visit many countries by car on weekend trips while a friend from high school became a Navy pilot and after his flight tours was able to work his way into a posting with NATO in Brussels.
As with all opportunities, it is what you make of it.
Best of luck