We study an information theoretic privacy mechanism design problem for two scenarios where the private data is either observable or hidden. In the hidden private data scenario, an agent observes useful data Y that is correlated with private data X , and generate disclosed data U which maximizes the revealed information about Y while satisfying a bounded privacy leakage constraint. Considering the other scenario, the agent has additional access to X . To design the privacy mechanism, we first extend the Functional Representation Lemma and Strong Functional Representation Lemma by relaxing the independence condition and thereby allowing a certain leakage. We then find lower and upper bounds on the privacy-utility trade-offs in both scenarios. In particular, for the case where no leakage is allowed and X is observable, our upper and lower bounds improve previous bounds. Considering bounded mutual information as privacy constraint and the observable private data scenario we show that if the common information and mutual information between X and Y are equal, then the attained upper bound is tight. Finally, the privacy-utility trade-off with prioritized private data is studied where part of X is more private than the remaining part.