The problem with a distributed storage system would not so much be joining the group and requesting data. That would be comparable to what bandwidth is used for services like bittorrent/dropbox/onedrive/google drive/... today. And the bandwidth usage of those services is small compared to media streaming services like netflix/youtube/twitch/... .
That being said, maintaining high availability and reliability of the data may be the bigger issue. If you can store data mainly on 'always-on' devices this is fairly simple, you choose a few group members, shard and replicate the data and it can live there for a long time. Only when a member dies you choose a new replication target.
However, most end-user devices with realistic app-install and storage capacity (i.e. not fridges) are laptops, desktops, phones, tablets, game consoles, NAS, ... . Only a few of these are always-on, most have the annoying habit of being switched off or losing connectivity on a regular basis. If this is the main bulk of your storage group/swarm, you need a fairly high replication rate to cover a multi-device loss and every time this rate dips too low because of many disconnect you need new replications. If this operations becomes frequent you may have huge bandwidth usage just to maintain the swarm.
So in this universe, if the company can gets lots of storage on always-on-devices the answer is likely no, the traffic will not be extraordinary. If they however mainly rely on more volatile devices the answer is most likely yes. Since at that point in the show they were targeting phones I don't think this would have worked very well, but we never found out.
But who knows, maybe aside from magical lossless compression algorithms, the silicon valley universe also has magical infinite (wireless) bandwidth :).