No. [I was wrong. In addition to being distributed between servers like I said, you can also enable P2P sharing to distribute the bandwidth even further.]
If you have a server that allows users to sign up, the stuff they follow/watch (you'd have to look at details if you want to host to see exactly how it's distributed) goes through your server.
The flip side to this is that, when your user uploads an extremely popular video (or you personally do if you don't allow signups), you don't have to stream every video to every individual user. You send it on to other federated instances that those users are signed up to, but if one instance has 100 users view your video, you don't have to send it 100 times. (This is likely less efficient than YouTube, because they can control exactly how load is spread between their delivery network with a comprehensive view of everything, but it dramatically lowers the barrier to entry for an individual to get involved or handle the distribution demand of a popular video.)
Just as a client, you don't serve anyone else. It's a website (or app) that works much like YouTube does. It's on the server side where the load is distributed.