  We discuss a left-right (L-R) symmetric model with the double seesaw
mechanism at the TeV scale generating Majorana masses for the active
left-handed (LH) flavour neutrinos $\nu_{\alpha L}$ and the heavy right-handed
(RH) neutrinos $N_{\beta R}$, $\alpha,\beta = e,\mu,\tau$, which in turn
mediate lepton number violating processes, including neutrinoless double beta
decay. The Higgs sector is composed of two Higgs doublets $H_L$, $H_R$, and a
bi-doublet $\Phi$. The fermion sector has the usual for the L-R symmetric
models quarks and leptons, along with three $SU(2)$ singlet fermion $S_{\gamma
L}$. The choice of bare Majorana mass term for these sterile fermions induces
large Majorana masses for the heavy RH neutrinos leading to two sets of heavy
Majorana particles $N_j$ and $S_k$, $j,k=1,2,3$, with masses $m_{N_j} \ll
m_{S_k}$. Working with a specific version of the model in which the
$\nu_{\alpha L} - N_{\beta R}$ and the $N_{\beta R} - S_{\gamma L}$ Dirac mass
terms are diagonal, and assuming that $m_{N_j} \sim (1 - 1000)$ GeV and ${\rm
max}(m_{S_k}) \sim (1 - 10)$ TeV, $m_{N_j} \ll m_{S_k}$, we study in detail the
new ``non-standard'' contributions to the $0\nu\beta\beta$ decay amplitude and
half-life arising due to the exchange of virtual $N_j$ and $S_k$. We find that
in both cases of NO and IO light neutrino mass spectra, these contributions are
strongly enhanced and are dominant at relatively small values of the lightest
neutrino mass $m_{1(3)} \sim (10^{-4} - 10^{-2})$ eV over the light Majorana
neutrino exchange contribution. In large part of the parameter space, the
predictions of the model for the $0\nu\beta\beta$ decay generalised effective
Majorana mass and half-life are within the sensitivity range of the planned
next generation of neutrinoless double beta decay experiments LEGEND-200
(LEGEND-1000), nEXO, KamlAND-Zen-II, CUPID, NEXT-HD.
