Mobile sites are incredibly important these days but some SEOs and webmasters are concerned that it will cause duplicate content issues.
Going responsive or the same URL, for the most part, removes the duplicate content worries completely. But some use different URLs or domains, such as an M DOT approach.
Google's John Mueller said in a Google Webmaster Help thread, "you don't have to worry about duplicate content -- definitely not when it comes to mobile versions when they're annotated appropriately."
He said this in response to some concerns addressed in the thread around duplicate content issues and mobile sites. Here is John's response:
You don't have to worry about duplicate content -- definitely not when it comes to mobile versions when they're annotated appropriately. In your case, you're using the same URLs anyway, so that wouldn't be an issue anyway. Serving different HTML for smartphone users is fine (we call it "dynamic serving" in our docs). Our main recommendation in this case is to use the "Vary" HTTP header so that the pages don't get cached in one or the other format (so that desktop users accidentally end up with the smartphone version, or vice-versa).
This is why I am a big advocate of the same URL and same content approach. But it doesn't work for everyone.
Forum discussion at Google Webmaster Help.