Not quite no, otherwise the load wouldn't be shared. How it works is that when you subscribe to a community (or instance) from that point on any comment, post or interaction is sent to the server you are on, and vice-versa. That way each server serves their users and the servers swap interactions with eachother to allow a certain level of scaling.
Now Lemmy I think doesn't download files (images etc), but points to the original instance, so it's really just the actual posts and comments (which will gradually add up over time). But Kbin, downloads the images and media too. That way, they take the download pressure too.
I've been running for less than a week with around 50 communities. The media storage is up to 5.5GB. Adding whole instances I'm pretty sure would increase the rate it rises quite a bit.