Else this is a privacy nightmare waiting to happen: It's easy to accidentially share numbers. Also it might turn out your new friend J. is superficially pleasant, but actually a horrible human being once you get to know them - congratulations, J. has access to all your shared contacts. If I was friends with Alice, I'd be very annoyed that J. is now harassing me.
Or suppose you created a list of all doctor's phone numbers in your area. You can then look up if any of your contacts (friends, family, loose aquintance) share any of them (e.g. because they recommended a specialist to a friend).
This needs a tremendous work to setup access rights, and it's easy for an individual to screw this up and put themselves in a bad spot.
Sharing contacts on a need-to-know basis (using e.g. Threema or WhatsApp) is the established standard in my peer group; be it the number of a Pizza place or my SO's new number, so a common friend can invite her to a party. And while it's less cool and adds some effort, it requires an explicit action to share a contact. This is more safe.
In my humble opinion, and I might have a bad hunch there: The overall mistake with that example is lies in how it associates identities with phone numbers. In reality, it's the other way around. We want to share identities (be it the Pizza Place, Mum, or a doctor), and should have control who can access the information (not just phone numbers) associated with them. But that's essentially re-inventing social networks with a narrow focus.
From a academical perspective the technical issue can be interesting of course, and valuable insights might be gained from this research. There might be use cases beyond the basic contacts example; in fact, I think in a company phone directory this can be incredibly useful as part of the CRM.