Liability for uploading video to a web site

If a website User uploads video, onto the website, of someone who agreed (verbally) to be filmed (and also was told that the video will be displayed online), do you think the website could be held liable for publishing the video if, prior to uploading, the User had agreed to the website Terms that essentially state “user will not upload content that is not lawfully users to upload to the website”?

It seems to me the website would always first be asked to remove the footage, prior to any legal problems. What do you think?

If the site is run/hosted in the US, then as a service provider you should generally be covered by the DMCA. But I believe you need a designated and registered DMCA agent, and you have to ensure you are not directly encouraging uploading of infringing content. So if there’s any problems, you should get a DMCA notice from the copyright holder, you then pull the content and contact the uploader to inform them, and if they insist they have the rights to upload the content, they can file a counter claim to have the content put back up.