Mechanism design for fully strategic agents typically assumes that agents broadcast their messages to a central authority that performs allocation and taxes/subsidizes agents. In addition, most of the designed mechanisms do not provide any guarantees that the Nash equilibrium (NE) will be reached asymptotically. Whenever such results are provided, they refer to convergence of specific predesigned learning dynamics and it is assumed that agents will follow such prescription. Both assumptions are an impediment in practical application of mechanism design in real-life allocation problems. In this article, we consider two common resource allocation problems: sharing K infinitely divisible resources among strategic agents for their private consumption (private goods), and determining the level for an infinitely divisible public good with P features, that is shared between strategic agents. For both cases, we present a distributed mechanism for a set of agents who communicate through a given network. In a distributed mechanism, agents' messages are not broadcasted to all other agents as in the standard mechanism design framework, but are exchanged only in the local neighborhood of each agent. The presented mechanisms produce a unique NE, they fully implement the social welfare maximizing allocation, and are budget-balanced at NE. Subsequently, it is shown that the mechanisms induce a game with contractive best-response, leading to guaranteed convergence for all learning dynamics within the adaptive best-response dynamics class, including dynamics such as Cournot best-response, k-period best-response, and Fictitious Play. While the abovementioned mechanisms have message dimensionality that grows quadratically with the number of agents, we show that if one is willing to forgo the strong convergence guarantees, it is possible to design a distributed mechanism with total message space dimensionality growing only linearly with the number of agents. Finally, we demonstrate the abovementioned results through a numerical study of convergence under repeated play, for various communication graphs and learning dynamics.