Note also that sharing does not imply "for free". It simply implies more than one person using a given resource.
I get that there's a real distinction between these new companies that facilitate peer-to-peer transactions. But I'd much prefer a term like "peer economy" than "sharing".
Sharing your car or home (whether the latter is rented or owned) is very easy without apps and people have been doing it for just about as long as cars and homes have existed. What the apps facilitate is making money from arms-length rentals to strangers. This may be a form of sharing, but even if so its a fairly specific one with its own name, its just that using an accurate and specific label rather than a misleadingly general one doesn't make as good marketing copy -- "sharing" sounds warm and friendly and positive, "rental" not so much.
Obviously people aren't sharing these things for free, but they are sharing them nonetheless.
There's a well-established term for the thing you call "go into business sharing" and its called "renting".