With composite $n_1$ = $p_1q_1$, and a separate $n_2 = p_1q_2$, can the primes be calculated more efficiently than factorization? Supposing that the (3 total) primes are kept secret? Does the reuse of $p_1$ allow an attacker to compromise $n_1$ and $n_2$ if the attacker guesses that both were generated with a shared prime between them (each having one unique prime)?
Yes, this trivially compromises them. Simply compute the gcd of $n_1$ and $n_2$, which will return $p_1$ (assuming $q_1 \neq q_2$). The gcd can be computed efficiently using Euclid's algorithm.