This definitely resonates with me and my peace corps service experience. The ability to communicate in not only in their target language, but also in their native language conveyed a since of, "if I can learn it, you can too."
3-9 months seems long for the average American in terms of travel, but it really isn't long enough to gain a social foothold and enough understanding in the local language to feel a certain connectedness to the place. I absolutely think any travel abroad does broaden one's horizons, but perhaps not to the extent that it's touted in pop culture.
I served in the US Peace Corps for two years, and the experience utterly shattered the person I was before, and re-formed me into someone far more understanding, compassionate, empathetic, etc. But that type of experience is highly focused with a great emphasis on cultural integration and language-learning.