A distributed control algorithm is proposed in this paper to achieve the leader-follower cooperative attitude tracking of multiple rigid bodies without using absolute and relative angular velocity information. The nonlinear manifold SO(3) is applied to represent the attitude of each rigid body. By considering the case of limited information exchange, i.e., not all the rigid bodies have access to the leader's information, a novel continuous nonlinear distributed estimator satisfying the rotation matrix dynamics is first introduced for each rigid body to estimate the leader's information asymptotically. Then, two auxiliary systems driven by proper inputs are designed such that the issues due to the angular velocity unavailability are effectively solved. Next, a feasible distributed protocol is designed by implementing the outputs from the distributed estimator and the auxiliary systems. It is demonstrated in terms of Lyapunov theory that, the developed distributed control algorithm ensures the asymptotically stable cooperative attitude tracking of multiple rigid bodies under the weak tree connectivity assumption. Finally, illustrative examples are provided to validate and highlight the proposed results.