Many biological processes involve the action of molecular motors that interact with the cell cytoskeleton. Some processes, such as the transport of cargoes, are achieved mainly by the action of individual motors. Others, such as cell motility and division, require the cooperative work of many motors. Collective motor dynamics can be quite complex and unexpected. One beautiful example is the bidirectional ("back and forth") motion of filaments which is induced when the motors within a group exert forces in opposite directions. In the talk, I will tackle the puzzle emerging from a recent experimental work in which it has been shown that the characteristic reversal times of the bidirectional motion are practically independent of the number of motors. This result is in a striking contradiction with existing theoretical models that predict an exponential growth of the reversal times with the size of the system. We argue that the solution to this puzzle may be the crosstalk between the motors which is mediated by the elastic tensile stress that develops in the cytoskeleton track. The crosstalk does not directly correlate the attachment and detachment of the motors, which work independently of each other. However, it highly accelerates their detachments by making the detachment rates system size dependent. Finally, I will demonstrate how the elasticity-mediated crosstalk influences the contraction of skeletal muscles, and provide an interesting explanation for the remarkable similarity in the sizes of sarcomeres in cell muscles of vertebrates.