Social media sites may or may not be public. If the user has privacy settings that do not open the page to the public, there are some privacy protections that can still apply. Furthermore, public or private, there are laws that can result in liability for posting certain content on your page.
The first rule to remember is that if the picture was taken of someone who had an expectation of privacy because of the place where the picture was taken, or the nature of the photo, then it may be an invasion of privacy for you to publish or republish the photo.
The second basic rule is that it is never okay to use someone's name or likeness in a commercial manner without a release from the subject. Commercial use would be either selling the image, or using the image to promote or sell something. If your social media site is being used by you in part to maintain a business related network, you could have a problem.
I hope this helps. Good luck to you.
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"Public domain" is a term of art in Copyright Law. Neither MySpace, nor Facebook are considered "public domain." Things that are considered public domain have no copyright protection. For example, the book "The Odyssey" by Homer is in the public domain. Please DO NOT assume that because something is made public, it is therefore "public domain." That would be wildly inaccurate.
The second part of your question involves the tort of invasion of privacy by way of posting a photo taken by somebody else, taken in their house. Without more facts, it is not possible to adequately answer this question. However, it is probably best to remove photos that you feel could offend another person, or, as you put it, invade their privacy.