Tailscale [0] says the private keys never leave the device.
“Security
Tailscale and WireGuard offer identical point-to-point traffic encryption.
Using Tailscale introduces a dependency on Tailscale’s security. Using WireGuard directly does not. It is important to note that a device’s private key never leaves the device and thus Tailscale cannot decrypt network traffic. Our client code is open source, so you can confirm that yourself.”