Really interesting thread! Many good considerations have been raised already. I do think it's important to understand the purpose of the clothing before snapping it up and putting it on--imagine a non-Westerner who shows up at a formal Western occasion in shorts and flip-flops, because shorts and flip-flops are to her "Western" clothing and thus all she needs to know in terms of matching the clothing with the event. I don't know if a Westerner would find her action culturally offensive, but she would definitely be rude in the sense of wearing inappropriate clothing to the formal event. On the other hand, think how many threads we have here debating whether X clothing is appropriate for Y event, just among Western clothing and events. You can learn about and avoid extremes, but there will always be grey areas in the middle where not everyone from a culture will agree.
I live in a multicultural city, and I don't think I've ever seen anyone in a full sari except those who could be ethnically Indian. I think they would get a lot of stares. What I think is interesting about cultural exchange is that it's so dynamic--in thirty years maybe lots of non-Indian people will be wearing saris and it won't seem weird at all. When my parents were young no non-Chinese people ate Chinese food on a regular basis, and now the Chinese place in my hometown is one of their favorite places to eat at.
For me personally, I don't think I would go with a full sari just out and about, but maybe part of one if that's possible, or something like a scarf or shawl made from similar fabric. Sometimes you do have a big culture leap when, say, a famous person wears something and it immediately becomes trendy and then normal for others to wear it; but I think more often it starts small, with jewelry or a bag, then moves up to a blouse or shoes, and eventually it doesn't seem odd for someone to be wearing an entire outfit that didn't originate with their culture, but piece by piece it's moved in. I think it's a really fascinating thing to study.