Common Modulus Attack not reproducible
In real word RSA modules are so large that probability for finding $c_1$ which is not coprime with $n$ is approximately zero.


  Also if you founded such number then $p=gcd(c_1,n)\neq1$ so $p$ is a factor of $n$ and in this case attack is not necessary because $n$ is factored.


$gcd(296,1073)=37\neq 1$ so $p=37,q=\frac{1073}{37}=29$ and $\phi(n)=1008$

Now you can easily compute private key $d_2$:

$e_2\cdot d_2=1 \pmod{ \phi(n)}$ so $d_2=275$.


  $$m={c_2}^{d2}\pmod n={555}^{275}\pmod{1073}=999$$