all data on nostr is public?

So are there no private relays? What if i wanted to create one for just me and my wife to use? Everything will go public because of gossip?

Doesn't this mean every relay will end up a copy of every post. Won't that make the data requirements for hosting a relay prohibitively expensive for most, thus centralising relays into the hands of big tech firms in the long term if Nostr does become popular?