Sample-to-sample free-energy fluctuations in spin-glasses display a markedly different behaviour in finite-dimensional and fully connected models, namely Gaussian versus non-Gaussian. Spin-glass models defined on various types of random graphs are in an intermediate situation between these two classes of models and we investigate whether the nature of their free-energy fluctuations is Gaussian or not. It has been argued that Gaussian behaviour is present whenever the interactions are locally non-homogeneous, i.e. in most cases with the notable exception of models with fixed connectivity and random couplings . We confirm these expectations by means of various analytical results concerning the large deviations of the free energy. In particular we unveil the connection between the spatial fluctuations of the populations of fields defined at different sites of the lattice and the Gaussian nature of the free-energy fluctuations. In contrast, on locally homogeneous lattices the populations do not fluctuate over the sites and as a consequence the small deviations of the free energy are non-Gaussian and scale as in the Sherrington–Kirkpatrick model.