When storytelling across cultures, we need to take into consideration several things: meaning, context, cultural gaps, important or difficult words/phrases and expressions, worldview, the plot/theme and intended meaning of the story, idiomatic expressions, historical implications, implicit information, frame of reference and shared frame of reference, etc. In short, we need to do a lot of homework! [All of this kind of takes the fun out of just telling a story, doesn't it?] Anyway, cultural anthropologists repeatedly say that a culture's stories reveal worldview. And I strongly believe this. I also believe that it is stories that shape worldview. Therefore, if one is not an "insider' of that culture, sometimes it is hard to understand its stories and tell its stories. The issue of language is really an important one. If you don't know a language well, you can't really tell a story well in that language. Word choices, implicit information, grammatical structures, idiomatic expressions, etc, all come into play here, and when telling through a translator, all of these things need to be worked through with the translator so the story can be conveyed with the meaning you intended but through understandable expressions. You can tell in a local language, and sometimes there's a certain charm to it. This, however, should become part of the storytelling "style" - playing on it to the storytellers advantage - but even in the broken simple language (or I should say especially in the broken simple language) meaning must be conveyed in understandable forms.
In cross-cultural situations, it is important that the storyteller conveys to the audience that he or she understands the audience and its culture, at least somewhat. Then, the stories in some way need to touch on that audience and its culture -- telling within that shared context or shared frame of knowledge, experience, and meaning. The audience appreciates it when it can relate to your stories, when they don't seem so foreign that the audience can't relate to them. If "foreign" stories are told to illustrate, inform or enlighten, they really must be told in ways that they can be understood and appreciated! So, it is good to tell some of "their" stories and even how, being an outsider, you messed up in sometimes telling their stories. Sometimes audiences enjoy some of the cross-cultural "snafus"! Oftentimes, they also want to be exposed to new cultures (that's why they come to hear "you"). But it is important to help them understand and not be confused! Be magical, be mystical, be enchanting, be entertaining, be informative. And be YOU. You and who you are are the greatest translator from one culture (yours) to another.