Can $n=pq$ be part of two different pairs of RSA keys?
If such keys exist, say $(e_1,n)$ and $(e_2,n)$, how are they related? What will be the security concerns for the two users?
The users will be able to read each other's messages (even though they can have different private keys, say $d_1$ and $d_2$). This is because knowledge of $d_i$ is sufficient to factor $N$, thus allowing that party to compute the other party's private key. This was detailed by Boneh in his analysis of RSA attacks.
Two properties of RSA are important here:
This means if you know one private key for a given $n$, you know all of them. Thus different persons should not share a modulus.
Such a scheme can be useful if one person needs multiple public keys. This person can use multiple $e$s with a shared $n$. There are few restrictions on the choice of $e$s, for example $ e_3 = e_1 * e_2 $ is broken.
Such key pairs are only useful in very specific scenarios. For example some blind signature schemes use different values of $e$ with shared $n$ to sign different currency denominations.