Collective cell migration, whereby cells adhere to form multi-cellular clusters that move as a single entity, play an important role in numerous biological processes, such as during development and cancer progression. Recent experimental work focused on migration of one-dimensional cellular clusters, confined to move along adhesive lanes, as a simple geometry in which to systematically study this complex system. One-dimensional migration also arises in the body when cells migrate along blood vessels, axonal projections and narrow cavities between tissues. We explore here the modes of one-dimensional migration of cellular clusters ("trains"), by implementing cell-cell interactions in a model of cell migration that contains a mechanism for spontaneous cell polarization. We go beyond simple phenomenological models of the cells as self-propelled particles, by having the internal polarization of each cell depend on its interactions with the neighboring cells, that directly affect the actin polymerization activity at the cell's leading edges. Both Contact Inhibition of Locomotion (CIL) and Cryptic Lamellipodia (CL) interactions between neighboring cells are introduced. We find that this model predicts multiple motility modes of the cell trains, that can have several different speeds for the same polarization pattern. Comparing to experimental data we find that MDCK cells are poised along the transition region where CIL and CL roughly balance each other, where collective migration speed is most sensitive to the values of the cell-cell interaction strength.