Ownership of private keys in the Lightning Network My question refers to the technical side: 

who is holding the private keys if a third party is using my lightning node to participate in the Lightning Network payment channel ? 

Thank you
If the third party is routing a payment through your node, only your node and its peers along the payment route need to sign, and therefore only the respective private keys are required. The third party node only signs with its private keys for channel states in its own channel it is routing through (the first hop in the payment route).

If the third party is accessing your node through the server API, then it is necessarily using your node wallet, so it is the private keys hosted on your node which are being used for channel state signing.

Private keys must generally be hosted on an active lightning node, as channel states are updated continuously by the implementation, without user intervention.