Differential privacy is a de facto privacy framework that has seen adoption in practice via a number of mature software platforms. Implementation of differentially private (DP) mechanisms has to be done carefully to ensure end-to-end security guarantees. In this paper we study two implementation flaws in the noise generation commonly used in DP systems. First we examine the Gaussian mechanism's susceptibility to a floating-point representation attack. The premise of this first vulnerability is similar to the one carried out by Mironov in 2011 against the Laplace mechanism. Our experiments show attack's success against DP algorithms, including deep learning models trained using differentially-private stochastic gradient descent. In the second part of the paper we study discrete counterparts of the Laplace and Gaussian mechanisms that were previously proposed to alleviate the shortcomings of floating-point representation of real numbers. We show that such implementations unfortunately suffer from another side channel: a novel timing attack. An observer that can measure the time to draw (discrete) Laplace or Gaussian noise can predict the noise magnitude, which can then be used to recover sensitive attributes. This attack invalidates differential privacy guarantees of systems implementing such mechanisms. We demonstrate that several commonly used, state-of-the-art implementations of differential privacy are susceptible to these attacks. We report success rates up to 92.56% for floating-point attacks on DP-SGD, and up to 99.65% for end-to-end timing attacks on private sum protected with discrete Laplace. Finally, we evaluate and suggest partial mitigations.