For proper implementation of digital contact tracing technologies for fighting against SARS-CoV-2, participants' privacy vulnerability and the uncertainty from the relevant institutions' side could be seen as two core elements that should be dealt with, among others. In this paper, we propose to understand the current approaches for preserving privacy, referred to as privacy by legislation and privacy by technological design, as distrusting strategies that primarily work to reduce participants' vulnerability by specifying and implementing privacy standards related to this digital solution. We point out that mere distrusting strategies are insufficient for the ethically appropriate development of this digital solution, nor can they eliminate the need for institutional trust that plays an essential role in fostering voluntary support for this solution. To reach well-grounded trust in both an ethical and epistemological sense, we argue that trust in institutions concerning personal data protection in the case of digital contact tracing ought to be built on the relevant institutions' and individuals' goodwill towards the public and their competence in improving the actual effectiveness of this solution. We conclude by clarifying three dimensions, including the purpose, procedure, and outcome, where the relevant trustees can work to signal and justify their intentions and increase their trustworthiness via an effective communication strategy. Given the complementary qualities shown by the distrusting and trusting strategies, a combined strategy including both sorts seems closer to what we expect from the responsible implementation of this digital solution, which could also improve the effectiveness of this institutional response.