Some people don't relapse once they've recovered, so they can be trusted
Recovered addicts that don't relapse are trustworthy people. One of the aspects of a recovered addict that makes them trustworthy is that they do not want to unravel the progress they've made. As a result, there is an incentive for recovered addicts not to relapse. To do this, recovered addicts will ensure that they continue their routines and/or treatments that prevent them from relapsing. Achieving this frame of mind takes effort and varies from person to person. For some people, it takes a substantial change or event for it to take place. The case of Pastor Robby Gallaty is one such example, who was able to beat his addiction after going to rehab and converting to Christianity.  An argument can be made that in addition to realizing that Gallaty needed to change and to keep that change in his life, finding religion helped reinforce his desire to prevent another relapse from occurring. His faith in Christianity serves him by keeping him honest, in turn making him trustworthy. Cases like these, however, are not universal. Despite being recovered, there is always the chance that a person can relapse. It depends on the person since some argue that addiction is a spectrum disorder.  It's nature as a spectrum disorder makes it difficult to gauge who is potentially trustworthy and who isn't. At the same time, it does not mean that recovered addicts that are trustworthy don't exist.
Counter to the idea that recovered addicts are trustworthy because they don't relapse, it may not be possible for an addict never to relapse again. The rationale behind this idea is that addiction is like a disease the addict has no control over. If the addict has no control over his/her disease, then they cannot control if and when they relapse.  It then becomes a matter of whether or not they can manage their addiction effectively.