Well, both sides of (1) are holomorphic functions of $\sigma$ in the strip $0<\Re(\sigma)<1$, because the integrand is holomorphic and rapidly decaying as $x\to\infty$ and the right hand side is holomorphic even at $\sigma=1/2$. So, yes: if (1) holds for $0<\sigma<1$ then it holds for $0<\Re(\sigma)<1$ in general, by the unicity theorem.

Well, both sides of (1) are holomorphic as functions of $\sigma$ in the strip $0<\Re(\sigma)<1$, because the integrand is holomorphic and rapidly decaying as $x\to\infty$ and the right hand side is holomorphic even at $\sigma=1/2$. So, yes: if (1) holds for $0<\sigma<1$ then it holds for $0<\Re(\sigma)<1$ in general, by the unicity theorem.