A search for heavy right-handed Majorana or Dirac neutrinos $NR$ and heavy right-handed gauge bosons $WR$ is performed in events with a pair of energetic electrons or muons, with the same or opposite electric charge, and two energetic jets. The events are selected from $pp$ collision data with an integrated luminosity of 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ collected by the ATLAS detector at $s$ = 13 TeV. No significant deviations from the Standard Model are observed. The results are interpreted within the theoretical framework of a left-right symmetric model and lower limits are set on masses in the heavy right-handed $W$ boson and neutrino mass plane. The excluded region extends to $mWR=4.7$ TeV for both Majorana and Dirac $NR$ neutrinos.