Connections between the electronic eigenstates and conductivity of one-dimensional disordered systems is studied in the framework of the tight-binding model. We show that for weak disorder only part of the states exhibit resonant transmission and contribute to the conductivity. The rest of the eigenvalues are not associated with peaks in transmission and the amplitudes of their wave functions do not exhibit a significant maxima within the sample. Moreover, unlike ordinary states, the lifetimes of these 'hidden' modes either remain constant or even decrease (depending on the coupling with the leads) as the disorder becomes stronger. In a wide range of the disorder strengths, the averaged ratio of the number of transmission peaks to the total number of the eigenstates is independent of the degree of disorder and is close to the value 2/5 , which was derived analytically in the weak-scattering approximation. These results are in perfect analogy to the spectral and transport properties of light in one-dimensional randomly inhomogeneous media [1], which provides strong grounds to believe that the existence of hidden, non-conducting modes is a general phenomenon inherent to 1D open random systems, and their fraction of the total density of states is the same for quantum particles and classical waves.