It appears that when the Meijer G function is discontinuous on the unit circle, the integrals over the left and the right loops can exist but differ. For $G_{1,1}^{0,1}\left(z\,\middle|\begin{array}{c}2\\0\end{array}\right)$, the integrand is $\frac{\Gamma(-1+s)}{\Gamma(1+s)}z^s=\frac{z^s}{s(s-1)}$. When $|z|=1$, the integral over $\mathcal L_-$ is $z-1$ and the integral over $\mathcal L_+$ is $0$.
How is the G-function defined in this case? Mathai and Saxena's Generalized Hypergeometric Functions omits the case $p=q, |z|=1$ in the definition and says the integrals always coincide provided they exist.