Efficient asynchronous Byzantine agreement (BA) protocols were mostly studied with private setups, e.g., pre-setup threshold cryptosystem. Challenges remain to reduce the large communication in the absence of such setups. Recently, Abraham et al. (PODC'21) presented the first asynchronous validated BA (VBA) with expected $O(n^3)$ messages and $O(1)$ rounds, relying on only public key infrastructure (PKI) setup, but the design still costs $O(λn^3 \log n)$ bits. Here $n$ is the number of parties, and $λ$ is a cryptographic security parameter. In this paper, we reduce the communication of private-setup free asynchronous BA to expected $O(λn^3)$ bits. At the core of our design, we give a systematic treatment of common randomness protocols in the asynchronous network, and proceed as: - We give an efficient reasonably fair common coin protocol in the asynchronous setting with only PKI setup. It costs only $O(λn^3)$ bits and $O(1)$ rounds, and ensures that with at least 1/3 probability, all honest parties can output a common bit that is as if randomly flipped. This directly renders more efficient private-setup free asynchronous binary agreement (ABA) with expected $O(λn^3)$ bits and $O(1)$ rounds. - Then, we lift our common coin to attain perfect agreement by using a single ABA. This gives us a reasonably fair random leader election protocol with expected $O(λn^3)$ communication and expected constant rounds. It is pluggable in all existing VBA protocols (e.g., Cachin et al., CRYPTO'01; Abraham et al., PODC'19; Lu et al., PODC'20) to remove the needed private setup or distributed key generation (DKG). As such, the communication of private-setup free VBA is reduced to expected $O(λn^3)$ bits while preserving fast termination in expected $O(1)$ rounds.