Quite simply, you can't access someone else's analytics data. Well, not without their username and password, or an extremely talented hacker, at least.
There are a few ways (think Alexa) you can estimate traffic - some of them are better than the rest, but all of them are imperfect. When you compare a website to another and use these metrics in a relative sense, however, they can be useful. Try looking up the estimated search volume of a competitors' brand names compared to your brand name using the Google Keyword Tool. This won't give you an accurate traffic estimate, but it might just help you figure out who's getting the most traffic and where your website falls in the market.