A space of boundary values is constructed for symmetric discrete Dirac operators in l(A)(2)(Z; C-2) (Z : {0, +/-1, +/-2, ...}) with defect index (1, 1) (in Weyl's limit-circle case at +/-infinity and limit-point case at +/-infinity). A description of all maximal dissipative (accretive), self-adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at +/-infinity. We investigate two classes of maximal dissipative operators with boundary conditions, called 'dissipative at -infinity' and 'dissipative at infinity'. In each of these cases we construct a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh-Weyl function of the self-adjoint operator. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative operators.