No.
Under Vimeo's Terms of Service, you are only permitted to upload videos for which you have all the necessary rights and permission, including any applicable music licenses.
This applies to all videos, regardless of privacy setting.
No matter what level of privacy restriction you choose for your videos, they will still be subject to the notice and takedown provisions of the Digital Millennium Copyright Act (“DMCA”).
Your best bet for avoiding strikes would be to review your uploads and fully delete any video that may infringe the copyright of a third party.