I think you're close -- its a critical mass issue in communities. There are countless networks to handle borrowing (some already mentioned) neighborrow, neighborgoods, peerby, nextdoor...the list goes on. There are also tool libraries in communities across the country and don't forget timebanks whose forums/email lists often turn to I'll borrow x for a time dollar.
Effectiveness is constrained by network adoption/critical mass -- Lyft isn't useful unless there are enough people to have someone available to pick you up when you need them.
I'm not sure any sort of app or website can replace that level of trust and sharing. Apps/websites don't replace the trust and sharing, they increase the effectiveness of your local relationships. In a lot of cases, living in the same area carries an amount of trust and sharing. Trusting relationships are actually built upon a certain level of risk that is taken at some point -- you trust your neighbor a lot more after you ask them to keep your key and an eye on your home while you're away for a weekend. So its sort of a chicken and egg question. Is there a lack in trust in local communities because neighbors/community members don't interact, therefore peer-to-peer tools will increase trust? Or will these types of networks not succeed because there isn't enough trust?