Some journals (example) today ask the authors to initiate a peer review rather than initiating a blind peer review themselves.
Are these articles different in terms of quality and bias? Are these articles accepted by experts/professors/scientists like regularly peer-reviewed articles?
Like the OP, I am in math and computer science. The situation described is not normal at all. For the vast majority of journals, the editor will choose the referee(s). Only once in my life did an editor ask me to recommend a referee, after the editor had failed to find anyone to referee the paper (that was for a CS paper submitted to a math journal, hence the difficulty).
It seems clear to me that allowing the author to choose the referee would subvert one of the main purposes of peer review. You could expect that papers would face less scrutiny on average, that more errors would get published, and that overall quality might suffer.
That said, because this practice is rare, probably most academics would not even know that the paper they are reading was peer reviewed in that way. I doubt it would truly affect whether or not others accept the paper as correct. Because we mostly communicate by preprints anyway, most readers already check papers for themselves, even if those papers have never been peer reviewed or published.
Still, all things being equal, as an author I'd recommend leaning away from journals where you pick the peer reviewer, because long term if that information gets out, that journal might be viewed with suspicion, and having published there might be held against you. Sure, it's tempting to publish something with less scrutiny, but we all know that publishing in vanity press can hurt you more than help you. The risk is not worth the reward. Best to stay with the normal publishing model, and only consider moving to new paradigms once you are already established.
