If we use the same algorithm to encrypt and sign a message, what can happen if the message is intercepted and read by a third party?
Although the names of the algorithms are the same, they are actually a little different when used for signing vs when used for encryption. The only thing similarity the two algorithms have is that they both use the same trapdoor permutation, conveniently called RSA. The difference is both in where certain variables come from, and the use of padding, which turns the insecure function into a cryptographically secure and useful algorithm. Encryption often uses RSA-OAEP, whereas signing uses RSA-PSS.
So even if you are using RSA/RSA for encryption and signing, an attacker who intercepts the message will be able to neither decrypt it nor forge valid ciphertext without first obtaining the relevant private key.