Plagiarism was a concern already in antiquity. In the preface to On Spirals Archimedes mentions that he was sending Alexandrians statements of his latest theorems, but some had claimed them as their own. Then on occasion he included two false ones "so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible".
The most famous case of ancient plagiarism however showcases the differences between then and now. The earliest documented concern about the data in Ptolemy's Almagest came from Kepler and Severin in 17th century, over a millennium after Ptolemy's death. Later Delambre in History of Astronomy (1817) pointed out that observations of solstices and equinoxes that Ptolemy reports as his own appear to be back-engineered from Hipparchus's data. In the 20th century thorough inspection of Almagest showed that there are the same problems with elongations of Venus and various other data that Ptolemy reports. The biggest problem is Ptolemy's star catalogue, analysis of which suggests that "his" observations were made 5° to the north of Alexandria, likely at Rhodes where Hipparchus worked.
In modern times this sustained a debate between "scientists" van der Waerden, Thurston, Rawlins and Richard Newton, and "historians" Neugebauer, Swerdlow and Gingerich on Ptolemy's conduct. Newton especially did not mince words in the tellingly named book The Crime of Claudius Ptolemy, concluding that he "is not the greatest astronomer of antiquity, but he is something still more unusual: he is the most successful fraud in the history of science". And Rawlins writes that "defenders invariably excuse Ptolemy's star thefts by noting that modern cataloguers commonly use other astronomers' stellar data. But such apologia just as invariably fail to mention that none of these borrowers claim such gleanings as their own outdoor labor". However, the defenders do have a point that some of this indignation comes from projecting modern standards onto antiquity, and that Ptolemy had to make something of sporadic and conflicting data without modern statistical methods. For more see debate in ISIS sparked by Thurston's Greek Mathematical Astronomy reconsidered.
Accusations of plagiarism continued into middle ages, Cardano for instance published solution to cubics given to him by Tartaglia after promising not to do so (he did give him credit though), and especially into 17th century, when Hooke alone made multiple plagiarism accusations, e.g. against Newton over the inverse square law, and even lobbied the Royal Society to stop sending its publications to non-members to prevent theft. The atrocious Newton-Leibniz priority dispute also had its share of plagiarism accusations from both sides. So already the first professional organizations, like the Academy of Sciences and the Royal Society, were very mindful of the plagiarism issue, and tried to address it from the start through institutional mechanisms. The main problem was distinguishing between true plagiarism, on which there was trivially a consensus, from "fair use" and uncertain timeline cases. For instance, the priority by publication date convention emerged after the Newton-Leibniz dispute to remedy the latter.