Copyright applies to images in the same way as to text. Whoever made the image retains ownership and chooses who is allowed to use the image and under what conditions. Photographs of human-made objects may be subject to additional constraints as they involve both the intellectual property of the designer of the object and that of the photographer; photographs of humans are also subject to additional constraints as they involve the subject's self-image rights.
Users are responsible for ensuring that they have the right to post the content that they contribute to Stack Exchange terms of service §3. Complaints of copyright violation must be sent to Stack Exchange's designated agent1 by the copyright holder or their designated agent — third parties are not competent to judge whether an image has been used by permission.
As a user, you may want to inform posters who reuse someone else's image that they may be in violation of copyright. Moderators will not handle copyright infringement complaints. This is outside their attributions. (If moderators handled copyright infringement complaints, that could make them legally responsible for any mistake that they make.)
Including an image inline is fundamentally different from linking. Linking is not regulated (however, linking to an image someone else's server while implying that the image is your own would be a copyright violation).
Plagiarism is a whole different matter from copyright. If someone posts an image without mentioning who the author of the image is, this is plagiarism. (It is also usually a copyright violation, but again this is not a point to pursue as a third party.) Plagiarism is forbidden on Stack Exchange.
If you notice plagiarism (of images or any other form of content), please do intervene. See What to do when plagiarism is discovered. In summary, if you can find the original author, edit the post to add an attribution. Sometimes the image wasn't improving the post by much; use your own judgement to decide whether to leave it in with attribution, link to the source instead, or remove it altogether. Leave a comment under the post linking to the help page “How to reference material written by others”. If the plagiarist rolls back an edit adding a required attribution, or in case of repeat or chronic offenders, flag for a moderator's attention and explain the situation. Moderators will delete posts and sanction users as appropriate.
My standard plagiarism comment is
[Plagiarism](/help/referencing) is [not cool](https://meta.stackexchange.com/a/134715).
- If the status of the DMCA "Service Provider/Designated Agent Information" contact linked above is no longer "Active", you should be able to find the correct contact by doing a search on dmca.copyright.gov for "Stack Exchange", "Stack Overflow", "stackoverflow.com", and/or "stackexchange.com".