You can follow anyone you like on Duolingo. You can also invite friends to join. When you follow someone, they will show up on your leaderboard, and you can compete against them. On the web, you can view your leaderboard by finding your left sidebar on your home page. You may need to scroll down to see it.
There, you can sort your friends by selecting the leaderboard of the week, month or all time. The highest scoring friends will be on top. This list is scrollable, so if you don't see the person you are looking for, try scrolling down while your mouse is on the leaderboard.
At the bottom of the leaderboard, there are a few buttons. You can find your Facebook friends by clicking on "Find more friends on Facebook."
You can also send someone an invite via email by clicking on "Send invite."
If your friend already uses Duolingo and you know their username or account email address, you can search for them within Duolingo.
You can also follow people you like on Duolingo. Simply click on their name in the forums or search for them. When you land on their profile page, click the "Follow" button to add them to your friend list. The person will automatically be added to your leaderboard and you will see updates of their activity on your activity stream. They do not have to accept your request, you can compete with anybody you like. They can also follow you back if they want to. If they block you, you can no longer add follow or contact them.