Electron beams escaping into interplanetary space produce faint hard X-ray emission along their coronal path in the so-called thin-target approximation at a level several orders of magnitude smaller than the main flare emissions. The only chance of observing this emission is in events that are well over the limb such that both the hard X-ray footpoints and the thermal X-ray sources are occulted. Theoretical calculations show that, under extremely favorable conditions, RHESSI observations could have enough sensitivity to detect thin-target emission from escaping electrons (Saint-Hilaire et al. 2008). Hard X-ray emission temporally correlated with radio type III bursts was observed by RHESSI from an elongated hard X-ray source in the corona, possibly outlining the electron escape path. However, the emission is about an order of magnitude too bright for purely thin-target emission from the number of escaping electrons seen near 1 AU by WIND/3DP. STIX will also not be able to regularly observe thin-target emission from escaping electrons, but, under favorable conditions such as a hard electron spectrum and high ambient plasma density, STIX will provide the first clean detections of purely thin-target emission from escaping electron beams.