First thought, naturally, is that an economy model based on money as the prime medium of an economic sphere will not create value by sharing. Luck, love or fun are things that grow when you share, money is not. If I share my Euro with you, I have less, of course, you have more, but it does not grow. But Certainly, there is only one way to gain money from sharing: If you provide an infrastructure for sharing which can be monetized. This is the second point – the microeconomic view from the point of a company or individual.
Most prominent example are ISPs (internet service providers). Napster, Gnutella or torrents were main drivers for growth in broadband internet contracts. Next prominent are sites like Facebook, YouTube, Twitter who “enable people to share” (Zuckerberg) – they also provide the infrastructure for sharing. Apart from the relationships and the people’s attention, here, it is not clear yet how to monetize the said infrastructure.
Still, there is a third way of regarding a share economy: A macroeconomic view that takes effects for the whole economy into account that are based on sharing. Maybe this is the most powerful effect, sharing can have on an economy, even if you only take the moneysphere into account. For economic growth, a society needs innovation. Innovation needs ideas and ideas never develop independently but are always a mashup of the existing in a new way. The more people share thoughts ideas, the more likely is the emergence of innovation.
So, to sum things up: Yes, there can be a share economy. Most likely if you do not measure in money but in other terms, like fun or joy. In the traditional, micro-economic way, most likely if you, as an organization provide an infrastructure for sharing. And third, on a macroeconomic level, for a society because sharing enables innovation and thus economic growth.