  We consider pure dephasing of Bell states of electron spin qubits interacting
with a sparse bath of nuclear spins. Using the newly developed two-qubit
generalization of cluster correlation expansion method, we calculate the spin
echo decay of $|\Psi\rangle$ and $|\Phi\rangle$ states for various interqubit
distances. Comparing the results with calculations in which dephasing of each
qubit is treated independently, we identify signatures of influence of common
part of the bath on the two qubits. At large interqubit distances, this common
part consists of many nuclei weakly coupled to both qubits, so that decoherence
caused by it can be modeled by considering multiple uncorrelated sources of
noise (clusters of nuclei), each of them weakly affecting the qubits.
Consequently, the resulting genuinely two-qubit contribution to decoherence can
be described as being caused by classical Gaussian noise. On the other hand,
for small interqubit distances the common part of the environment contains
clusters of spins that are strongly coupled to both qubits, and their
contribution to two-qubit dephasing has visibly non-Gaussian character. We show
that one van easily obtain information about non-Gaussianity of environmental
noise affecting the qubits from the comparison of dephasing of $|\Psi\rangle$
and $|\Phi\rangle$ Bell states. Numerical results are obtained for two nitrogen
vacancy centers interacting with a bath of $^{13}$C nuclei of natural
concentration, for which we obtain that Gaussian description of correlated part
of environmental noise starts to hold for centers separated by about 3 nm.
