RSA has this property, but the public and private keys are not chosen arbitrarily. The public key can easily be derived from the private key, but there is no obvious way to determine the private key from the public key.
Nitpick: in RSA implementations using the speed optimization of a small public exponent (sometimes even always the same small number, like 3 or 65537), the private exponent has to be derived from the public one. This derivation uses some secret information (the factors of the modulus), so not just anyone can do it. The standard private key file format includes the factors of the modulus to enable some speedups for private key operations; however, the math doesn't require this, and both public and private key operations could be done with just an exponent and a modulus. Someone with just a private exponent and modulus wouldn't be able to compute the matching public exponent, unless it were small and easy to guess.