LastPass and I'm sure many other password managers claim that they encrypt and decrypt the data ONLY on your local device and that master key is never sent to LastPass servers - https://www.lastpass.com/how-lastpass-works
My question is: How the password that's encrypted on my device using my own key can be decrypted by other people with whom I shared that entry with ?
They use public-key cryptography. Your device generates an RSA key pair. The private key is encrypted with your vault encryption key, and the public key is given to everyone who wants to share information with you. The information they encrypt with your public key can only be decrypted with your private key, which only you can access (since it is protected by your vault encryption key, which is protected by your password).
https://support.lastpass.com/download/lastpass-technical-whi...