We build a realizability model for linear logic using a name-passing process calculus. The construction is based on testing semantics for processes, drawing ideas from spatial and modal logics, and yields a new type system for process calculi that ensures termination while allowing significantly concurrent behaviours. Then we study how embeddings of intuitionistic and classical logics into linear logic induce typed translations of lambda and lambda-mu calculi in which new concurrent instructions can be introduced, thus sketching the basis for a Curry-Howard interpretation of linear and classical proofs in terms of concurrent interaction.