Actually great questions. Yes and no. There are vulnerabilities if the private key leaks, but public keys are just that; perfectly okay public in any hands. You only encrypt data with it.

What makes the Signal protocol so awesome, and other algorithms like it, is that it reduces the threat surface area further by using onetime keys. So even if your key is leaked, it cannot be used to decrypt old or forthcoming messages as the keys have already ratcheted to the next pair.