We study the local dynamical fluctuations in glass-forming models of particles embedded in d-dimensional space, in the mean-field limit of d→∞. Our analytical calculation reveals that single-particle observables, such as squared particle displacements, display divergent fluctuations around the dynamical (or mode-coupling) transition, due to the emergence of nontrivial correlations between displacements along different directions. This effect notably gives rise to a divergent non-Gaussian parameter, α2. The d→∞ local dynamics therefore becomes quite rich upon approaching the glass transition. The finite-d remnant of this phenomenon further provides a long sought-after, first-principle explanation for the growth of α2 around the glass transition that is not based on multiparticle correlations.