We propose a novel approach for solving the Automatic Step-size Adjustment problem in Newtown-Raphson Consensus based distributed optimization algorithms (abbreviated as ASAN). The approach leads the agents in the network to autonomously, continuously and automatically choose their local stepsizes as the distributed optimization process unfolds. In practice, it is based on first evaluating the reliability of the next predicted local optimum by using the information collected at each node as the optimization process is being executed, and then using this reliability assessment to locally adjust the step-size accordingly. By letting each node adjust its own step-size separately by means of local inspection of local variables, the approach does not add communication overheads (a feature that is beneficial especially for systems with limited communication possibilities). Moreover, the strategy does not require information about the current topology of the communication network, nor information about the local cost functions, serving thus situations for which both the network structure and the local cost functions are time-varying. Besides the overall concept, the paper introduces different heuristic reliability evaluation algorithms to analyse the temporal dynamics of the local data, and compares the approach against an oracle-based implementation that selects the best constant step-size for each specific network and set of local costs by means of Monte Carlo analyses. The paper statistically shows that the proposed heuristic leads to convergence rates that are most often better (and in general not worse) than the ones of the oracle-based optimization scheme, without though the need for knowing information about the communication topology or local costs.