Methods and systems are provided for operating a split exhaust engine system that provides blowthrough air and exhaust gas recirculation to an intake passage via a first exhaust manifold and exhaust gas to an exhaust passage via a second exhaust manifold. In one example, a first set of exhaust valves coupled to the first exhaust manifold may be operated at a different timing than a second set of exhaust valves coupled to the second exhaust manifold. Further, a position of a first valve positioned in a first passage coupled between the intake passage and the first exhaust manifold and/or a timing of the first set of exhaust valves may be diagnosed based on an output of a pressure sensor positioned in the first exhaust manifold.