  In this paper, we introduce the reverse-space and reverse-space-time nonlocal
discrete derivative nonlinear Schr\"odinger (DNLS) equations through the
nonlocal symmetry reductions of the semi-discrete Gerdjikov-Ivanov equation.
The muti-soliton solutions of two types of nonlocal discrete derivative
nonlinear Schr\"odinger equations are derived by means of the Hirota bilinear
method and reduction approach. We also investigate the dynamics of soliton
solutions and reveal the rich soliton structures in the reverse-space and
reverse-space-time nonlocal discrete DNLS equations. Our investigation shows
that the solitons of these nonlocal equations often breathe and periodically
collapse for some soliton parameters, but remain nonsingular for other range of
parameters.
