Local differential privacy is a strong notion of privacy in which the provider of the data guarantees privacy by perturbing the data with random noise. In the standard application of local differential differential privacy the distribution of the noise is constant and known by the learner. In this paper we generalize this approach by allowing the provider of the data to choose the distribution of the noise without disclosing any parameters of the distribution to the learner, under the constraint that the distribution is symmetrical. We consider this problem in the unconstrained Online Convex Optimization setting with noisy feedback. In this setting the learner receives the subgradient of a loss function, perturbed by noise, and aims to achieve sublinear regret with respect to some competitor, without constraints on the norm of the competitor. We derive the first algorithms that have adaptive regret bounds in this setting, i.e. our algorithms adapt to the unknown competitor norm, unknown noise, and unknown sum of the norms of the subgradients, matching state of the art bounds in all cases.