So when verifying the transaction using your private key will the private key be known or seen by others on the network? If so then is it not risky letting others know our private keys.
The private key should remain secret and you should never share it with anyone. That is an important part for safety of your cryptocurrency.
What you give out to people is your public address, which is derived from your private key and when you use your private key to sign transaction on the blockchain, it lets the network know that you signed it with your private key without you having to show the actual private key to anyone. That is a core thing of how asymmetric encryption works that makes cryptocurrency secure.
So remember, keep that private key private and dont show it to anybody no matter what.
And also in a case where the owner loses his/her private key and is unable to verify the transaction what will happen to the coins. Are they gone forever? Or is there a way to recover the private key.
The private key is not used to verify, but to cryptographically sign transactions. Without a private key, you will no longer be able to sign transactions, therefore you will not be able to spend coins from your address.