We present a local approach to the study of scale-to-scale energy transfers in magnetohydrodynamic (MHD) turbulence. This approach is based on performing local averages of the physical fields, which amounts to filtering scales smaller than some parameter ℓ. A key step in this work is the derivation of a local Kármán-Howarth-Monin relation which provides a local form of Politano and Pouquet's 4/3 law, without any assumption of homogeneity or isotropy. Our approach is exact and nonrandom, and we show its connection to the usual statistical results of turbulence. Its implementation on data obtained via a three-dimensional direct numerical simulation of the forced incompressible MHD equations from the John Hopkins turbulence database constitutes the main part of our study. First, we show that the local Kármán-Howarth-Monin relation holds well. The space statistics of local cross-scale transfers are studied next, their means and standard deviations being maximum at inertial scales and their probability density functions (PDFs) displaying very wide tails. Events constituting the tails of the PDFs are shown to form structures of strong transfers, either positive or negative, which can be observed over the whole available range of scales. As ℓ is decreased, these structures become more and more localized in space while contributing to an increasing fraction of the mean energy cascade rate. Finally, we highlight their quasi-one-dimensional (filamentlike) or quasi-two-dimensional (sheetlike or ribbonlike) nature and show that they appear in areas of strong vorticity or electric current density.