The pseudorapidity distributions of charged hadrons in proton-lead collisions at nucleon-nucleon center-of-mass energies root s(NN) = 5.02 and 8.16 TeV are presented. The measurements are based on data samples collected by the CMS experiment at the LHC. The number of primary charged hadrons produced in non-single-diffractive proton-lead collisions is determined in the pseudorapidity range vertical bar eta(lab)vertical bar < 2.4. The charged-hadron multiplicity distributions are compared to the predictions from theoretical calculations and Monte Carlo event generators. In the center-of-mass pseudorapidity range vertical bar eta(cm)vertical bar < 0.5, the average charged-hadron multiplicity densities < dN(ch)/d eta(cm)>vertical bar(vertical bar eta cm vertical bar) < 0.5 are 17.1 +/- 0.01 (stat) +/- 0.59 (syst) and 20.10 +/- 0.01 (stat) +/- 0.5(syst) at root s(NN) = 5.02 and 8.16 TeV, respectively. The particle densities per participant nucleon are compared to similar measurements in proton-proton, proton-nucleus, and nucleus-nucleus collisions.