Twitter has rolled out a nifty little feature. Now on anyone's profile, when you click on the "Following" link, you'll be able to see exactly how they see Twitter. In other words, you can see the same timeline of tweets that they see when they're looking at their main feed.
For example, check out what diTii's timeline looks like by going over here. As you can see, Twitter added the ability to see what the 418 people on Twitter that diTii follows are "tweeting right now."
Bonus, if you look at the upper right corner of the following page you'll see a "shuffle" icon. Hovering over this reveals the option to "jump to someone you follow". So you can jump between people you follow and find new people to follow from their streams.
This is a similar concept to what News.me does for news stories. There, you look at a person's profile and can see the shared links that they see in their Twitter stream. (It's also worth noting that Twitter actually had this feature back in the day but shut it off a number of years ago.)