We report observations of reconnection exhausts and associated ion and electron separatrix layers in and around the Heliospheric Current Sheet (HCS) during PSP Encounters 08 and 07, at 16 R_s and 20 R_s. HCS reconnection accelerated protons to almost twice the solar wind speed and increased the proton core energy by a factor of ~3, due to the Alfvén speed being comparable to the solar wind flow speed at these near-Sun distances. During E08, accelerated protons were found to have leaked out of the exhaust along separatrix field lines, appearing as field-aligned energetic proton beams in a broad region outside the HCS. Concurrent dropouts of strahl electrons, indicating disconnection from the Sun, provide further evidence for the HCS being the source of the beams. Around the HCS in E07, there were also proton beams but without electron strahl dropouts, indicating that their origin was non-local and closer to the Sun.