Technically, someone else's work, whether a photo, video, or writing belongs to them. They have copyright over it, whether they specifically say they do or not (in America anyway, other countries vary).
Some people specifically waive or modify their copyrights, such as through Creative Commons license, which lays out the ways in which it is acceptable to use their work. Some people will say go ahead and do anything you want with it. If there is no specific copyright information than one must assume it is under copyright. Using it without permission, unless you believe it to be justified under a fair use principle, is wrong.
Legality is a separate issue. It only really becomes illegal if you use someone's work and they then bring a suit against you, which they then would have to win. If you used the work under a fair use claim, they would have to also prevail against your defense. Fair use is a doctrine which allows limited use of a work without permission. Common examples include for education, news reporting, or criticism.
Sharing Youtube videos is generally probably safe, because most people I think assume that their videos may be shared (and in many cases want them to be). Also, many people posting videos on Youtube don't have the ability to go around suing people who use their videos without permission. That doesn't technically mean it is right. It is always best to get permission.