A study of forward energy flow and central charged-particle multiplicity in events with W and Z bosons decaying into leptons is presented. The analysis uses a sample of 7 TeV pp collisions, corresponding to an integrated luminosity of 36 pb(-1), recorded by the CMS experiment at the LHC. The observed forward energy depositions, their correlations, and the central charged-particle multiplicities are not well described by the available non-diffractive soft-hadron production models. A study of about 300 events with no significant energy deposited in one of the forward calorimeters, corresponding to a pseudorapidity gap of at least 1.9 units, is also presented. An indication for a diffractive component in these events comes from the observation that the majority of the charged leptons from the W(Z) decays are found in the hemisphere opposite to the gap. When fitting the signed lepton pseudorapidity distribution of these events with predicted distributions from an admixture of diffractive (POMPYT) and non-diffractive (PYTHIA) Monte Carlo simulations, the diffractive component is determined to be (50.0 +/- 9.3 (stat.) +/- 5.2 (syst.))%.