To "take" an incoming payment with the public channel and forward it through the private one, require some manual scripting and building the route. It is not done automatically.

Otherwise any private node with a single public channel could route. And that's not happening right now. Maybe some LN guru is doing that with complex scripts but is not usual for any software.

DarthCoin

lightning

Does the second channel have to be public for routing?

Shadow liquidity means that if you have both a public and a private channel to the same node, payments could be routed through the private channel, but this is contingent on the public channel being used to discover the route. In this case, the routing node would know of a path to the public channel, but it might send liquidity through the private one, providing a sort of hidden liquidity for routing.