We study the conformations of a self-avoiding polymer confined to a channel by computing the cross-sectional distributions of the positions of its monomers. By means of Monte Carlo simulations for a self-avoiding, freely jointed chain, we determine how the cross-sectional distribution for a given monomer depends on its location in the polymer and how strongly this distribution is affected by self-avoidance. To this end we analyze how the frequency of intrachain collisions between monomers depends on their spatial position in the channel and on their location within the polymer. We show that most collisions occur between closely neighboring monomers. As a consequence, the collision probability depends only weakly on the spatial position of the monomers. Our results explain why the effect of self-avoidance on the monomer distributions is weaker than predicted by mean-field theory. We discuss the relevance of our results for studies of DNA conformations in nanofluidic channels.