We evaluate the consequences of interactions between charge carriers on the transport dynamics or currents in normally insulating one-dimensional systems where excess charges have been injected electrically, chemically, or optically. The time-dependent currents and space charge distribution are calculated using a many body Monte Carlo method adapted to stochastic transport. The idea is to study collective interaction phenomena and to learn how to distinguish them from disorder effects. The latter can in principle be modeled to explain almost any experimental result, however, this is often done at the expense of having to use unphysical or inconsistent parameters. Interactions can we argue, give for example, rise to low frequency anomalies in ac conductivity, negative differential resistance, low field spin magnetoresistance, and critical nonlinear noise like oscillations in currents as a function of time.