Two possible variants of a video sharing site like YouTube:
1. After taking down a video in the execution of a DMCA order, the description remains, and the comments remain open, because those were not infringing copyright.
2. Upon receiving a DMCA takedown request, the request is published at a central location, and the video remains up for a grace period. Ostensibly, this allows supporters of the video beyond the uploader to gather and organize themselves to legally fight the DMCA order in court.
Either of these two mechanisms provide enough rope for creating and publicizing a link to an alternate site for the video, perhaps in a censorship resistant medium, which I suspect is why YouTube does not allow them. Are they forbidden by law? The general idea is not to foil the Streisand Effect from combating censorship.