  Open systems of coupled qubits are ubiquitous in quantum physics. Finding a
suitable master equation to describe their dynamics is therefore a crucial task
that must be addressed with utmost attention. In the recent past, many efforts
have been made toward the possibility of employing local master equations,
which compute the interaction with the environment neglecting the direct
coupling between the qubits, and for this reason may be easier to solve. Here,
we provide a detailed derivation of the Markovian master equation for two
coupled qubits interacting with common and separate baths, considering pure
dephasing as well as dissipation. Then, we explore the differences between the
local and global master equation, showing that they intrinsically depend on the
way we apply the secular approximation. Our results prove that the global
approach with partial secular approximation always provides the most accurate
choice for the master equation when Born-Markov approximations hold, even for
small inter-system coupling constants. Using different master equations we
compute the stationary heat current between two separate baths, the
entanglement dynamics generated by a common bath, and the emergence of
spontaneous synchronization, showing the importance of the accurate choice of
approach.
