There is no single answer to this question, as it depends for each text on what you have agreed with the publisher in writing. It is therefore vital that, before signing a contract, you carefully examine which rights of use you grant to the publisher and their scope (also refer to the FAQs provided by the Swiss National Science Foundation on this topic [PDF, sections 12 and 13 in particular]). However, the vast majority of publishers will allow you to make available either the post-print (author's last version) or the preprint (submitted but not yet peer-reviewed version) through a repository or your website. The SHERPA/ROMEO database will provide you with information on publisher policies.
Plagiarizing electronic resources with copy and paste is certainly easier than copying out information from a book. However, plagiarism was around before the Internet. And plagiarism doesn't just apply to Open Access literature but to any electronic text, whether it is Wikipedia or the online edition of a daily newspaper. However, as soon as the full text of the original document is online, it is also much easier to recognize plagiarism by comparing text!
In addition, Open Access publishers and journals mostly work with CreativeCommons licenses that safeguard authors' intellectual property rights and ensure their right to be named as the author. Furthermore, authors choose themselves how and to what extent their publication may be used by others (whether for non-commercial use only, whether it may be edited, etc.). This means that the authors – not the publisher – decide who is allowed to work with their texts, and how.