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Private channels aren't private. They should be called "unannounced channels".
They simply aren't included in your node's gossip. This means your node isn't telling all it's peer about the channel. The unannounced channel is excluded from the public network graph built by every node.
If Alice wants Charlie to use her unannounced channel with Bob, she can include a "route hint" in the invoice itself.
Then, anyone who sees that invoice knows Alice has an unannounced channel with Bob and can use it to route a payment on that channel
It's also possible to probe for unannounced channels and use them without any "permission" or route hint.
An unannounced channel is just a "suggestion" to the network that says, "hey, I'd prefer if none of you use this one."
But anyone who's determined enough can still use it.
Soon we will have short channel IDs (SCIDs) which will help hide unannounced channels better.
Yes, we have a side project. We just don't think the market is quite there or needs it at all. Because LN txns are low value, phones are perfectly good for it.