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Distributed clause-sharing SAT solvers can solve challenging problems hundreds of times faster than sequential SAT solvers by sharing derived information among multiple sequential solvers. Unlike sequential solvers, however, distributed solvers have not been able to produce proofs of unsatisfiability in a scalable manner, which limits their use in critical applications. In this work, we present a method to produce unsatisfiability proofs for distributed SAT solvers by combining the partial proofs produced by each sequential solver into a single, linear proof. We first describe a simple sequential algorithm and then present a fully distributed algorithm for proof composition, which is substantially more scalable and general than prior works. Our empirical evaluation with over 1500 solver threads shows that our distributed approach allows proof composition and checking within around 3x its own (highly competitive) solving time.