We analyze whether circuit QED Hamiltonians are stoquastic, focusing on systems of coupled flux qubits. We show that scalable sign-problem-free path integral Monte Carlo simulations can typically be performed for such systems. Despite this, we corroborate the recent finding [I. Ozfidan, Phys. Rev. Appl. 13, 034037 (2020)10.1103/PhysRevApplied.13.034037] that an effective, nonstoquastic qubit Hamiltonian can emerge in a system of capacitively coupled flux qubits. We find that if the capacitive coupling is sufficiently small, this nonstoquasticity of the effective qubit Hamiltonian can be avoided if we perform a canonical transformation prior to projecting onto an effective qubit Hamiltonian. Our results shed light on the power of circuit QED Hamiltonians for the use of quantum adiabatic computation and the subtlety of finding a representation which cures the sign problem in these systems.