We should distinguish between the peer review process as it exists today (in which academic publishers require a review — sometimes anonymous, sometimes not — of a work before it is accepted for publication) and between the general task of other scientists arguing with one another and checking their work. The latter goes back to any meaningful definition of "science," because the entire point is that you are engaging with theories, observations, evidence, arguments, etc. of other people.

To illustrate the difference, Darwin's Origin of Species was not peer-reviewed before publication — it was simply published. However his work is full of engagement with the work of other scientists, and his work was scrutinized, criticized, attacked, etc. by other investigators. He was participating at a time when there were many interested "peers" who were reading and critiquing his writing — but this is not peer review.

An early instance of academic peer review, or something like it, can be found in Henry Oldenburg's Philosophical Transactions in the 17th century. Oldenburg would regularly refer manuscripts to other experts to decide their quality. However peer review did not become a standard practice of journals until the late 19th century, when science itself began to truly "professionalize" and look more or less the way it does today. But a journal as prestigious as Nature routinely avoided peer review though the 1960s. (Watson and Crick's famous 1953 paper describing the structure of DNA was, for example, not peer-reviewed. A later editor of Nature suggested it would not have been accepted if it had — the evidence it offered up was pretty slim for their claims, and it was Rosalind Franklin's evidence anyway.) So it can be hard to generalize.

While popularly most people think peer review is what makes science good, there are many scientists (and social scientists who study scientists) who have pointed out its many, many deficiencies.

My information above about the history of peer review comes from my friend Melinda Baldwin's recent book, Making "Nature": The History of a Scientific Journal (University of Chicago Press, 2015).

If you are asking in general, when did people start making arguments about the natural world in a systematic way... this is a very controversial question. The typical answer is "the Scientific Revolution" (e.g. the 15th-18th centuries) but many historians of science do not really believe there was any distinct Scientific Revolution during this time, and if your bar is just people disagreeing with each other, you can find that much earlier than that. The institutions of what we today call "science" started in the 17th century or so, but did not really become complete (to whatever degree they are currently "complete") until the late 19th century. By the late 19th century, you can imagine someone saying, "ah, I'll be a scientist!" (the word came into use then) and people would have a good idea of what that meant, both in terms of what they were planning to become and the path that would lead to that career.