Legally, the default is that you need to ask permission of all subjects to shoot the photo. And again for each publication of that image.

In practice this is relaxed for public shots and street photography where the intent is not to make someone the subject of the photograph (people walking by in the background of a shot) or when "treating people with respect" (so yeah, no predatory homeless shots, or getting up in people's faces). That said, this is for personal use and publishing them means you still might need to seek permission or risk being prosecuted later.

Finally there is a caveat that is "the right for information" which is how the paparazzi are able to photograph celebrities and the like. Under the heading of journalism.

Even so, you can only use those photographs in context. So, say you catch a shot Madonna flouting the law, you can't later use that same photograph to illustrate an article of her being nice to cats. Furthermore they can also claim you are intruding on their private life. Which might still get you into trouble.