In this paper, we extend linear combinatorial bandits \cite{gai2012combinatorial} to a multiplayer setting with information asymmetry \cite{chang2022online, chang2024optimal}, where each player controls an arm and independently decides whether to pull it, with coordination allowed only before rounds begin. We analyze three scenarios: action asymmetry (players can't observe others' actions but receive identical rewards per iteration), reward asymmetry (players observe actions but receive private i.i.d.\ rewards), and combined asymmetry. We derive near-optimal, gap-independent regret bounds for all scenarios: For action or reward asymmetry, we achieve $\tilde{\mathcal{O}}(\sqrt{T})$, which improves significantly from \cite{gai2010learning}; for both action and reward asymmetry, we achieve near-optimal bounds similar to that of \cite{chang2022online}. We finally generalize our results to settings where players decide either not to pull or to pull one out of multiple arms, and achieve similar bounds in similar settings as above.