We consider a model of self-propelled agents with spring-like interactions that depend only on relative positions, and not on relative orientations. We observe that groups of these agents self-organize to achieve collective motion (CM) through a mechanism based on the cascading of self-propulsion energy towards lower elastic modes. By computing the correlation functions of the speed and velocity fluctuations for different group sizes, we show that the corresponding correlation lengths are proportional to the linear size of the group and have no intrinsic length scale. We argue that such scale-free correlations are a natural consequence of the position-based interactions and associated CM dynamics. We hypothesize that this effect, acting in the context of more complex realistic interactions, could be at the origin of the scale-free correlations measured experimentally in flocks of starlings, instead of the previously argued proximity to a critical regime.