The answer depends on your research question. If you were interested in strictly a one-shot test of whether a given path model works "satisfactorily" (and you'll have to define that criterion operationally for your study), then you may stop with the reporting and interpretation of the estimated path coefficients (and variance accounted for in endogenous variables). Model-data goodness of fit indices are usually helpful as well, if the model is not fully saturated.
On the other hand, if your research goal is to identify the most parsimonious model possible (and you discard paths that turn out to be not significantly different from zero), then you would want to re-estimate the model, and report on the original and subsequent path coefficients (and goodness-of-fit).
However, modification of the model after having looked at the results of the original model means that you have a much higher likelihood of having capitalized on characteristics that may be unique to your sample (and therefore might not generalize well to the population or to other samples), so replication is very important in this circumstance.