What should I do if I discover a paper I rejected because of methodological flaws has been published in another journal? I recently discovered that a paper I reviewed a while back and rejected as a result of major methodological flaws had been resubmitted to another journal and published without addressing any of the significant errors that required the paper to be rejected in the first place. Is there anything I can do that doesn't violate the confidentiality of the peer review process to address the problem?
Two things are unclear about your question Is there anything I can do that doesn't violate the confidentiality of the peer review process to address the problem?:


What is the goal of any intervention?
Why do you think that anything should be done?


If, for example, you aim for retraction, I guess this could be difficult. You may go for this on the ground that you can prove that the authors were aware of the flaws you pointed out and submitted the paper knowing that there was something wrong. The journal may have some rule that the authors have to certify that the results are true to the best of their knowledge. If this really is so, you may have some angle of attack. If you aim for correction by the authors, it could also be difficult. However, you could consider writing a letter to the editor, pointing out what is wrong and then ask if the journal would publish such a letter. Some journals I know have such formats. You could also do follow-up research on the same topic and write a paper yourself where you refer to the paper with flaws and describe what wrong there.

For the second question, it sounds like you feel that "What's published and peer-reviewed should be true." While this resonates with me, it's not something that is close to true now and probably never will be. Mistakes happen. You may then trust in the scientific community that the flawed paper will be perceived as such in the long run and find peace with this particular paper.

Finally, the premise that confidentiality of the peer review process may permit some actions is not totally clear. As I wrote in a comment, I asked a question about this on MO and got very different responses.