Why is it a security problem that two RSA moduli are not co-prime?
The public key is actually $(n,e)$ where $n = p \, q$: it's the modulus, which is the product of the two primes, that appears in the public key.

Suppose that you have two public keys $(n_A, e_A)$ and $(n_B, e_B)$ with $n_A \neq n_B$, and that $\gcd(n_A, n_B) \neq 1$. You have a common divisor of both $n_A$ and $n_B$. But $n_A$ and $n_B$ have only two divisors each, the two primes. Therefore $\gcd(n_A,n_B)$ is one of the secret primes of both keys, and the other secret primes are trivially $n_A / \gcd(n_A,n_B)$ and $n_B / \gcd(n_A,n_B)$.

This shows that it's critical that in an RSA key, both primes are unique.

Knowing a factor of the two totients may or may not reveal the private key. If the factor you have is $4$, that doesn't give you any information: $(p-1)(q-1)$ is always a multiple of $4$. If the factor you have happens to be $p-1$ or to let you find $p-1$, that's obviously a problem. But since the totient is not normally exposed anywhere, this isn't a common concern.