A private Facebook video cannot be downloaded through Pullvio because the URL does not contain your Facebook viewing permission. Pullvio accepts public video, Reel, Watch, share, and fb.watch links; it does not ask for account cookies or sign in as you. If the post only plays inside your logged-in session, it is not a public source.

A link identifies the post, not its audience permission
Facebook evaluates the viewer as well as the URL. The same address can play for the uploader, friends, group members, or an eligible signed-in user while showing nothing to a signed-out visitor. Copying it does not turn the audience setting into Public.
Meta's help center says people who are not logged in can see information shared with the Public audience. It also notes that only the uploader can change a video's audience. That is why asking a downloader to “use the private link” is not a technical repair.
Use the signed-out test
Open a private browser window and paste the Facebook URL. If the player is visible without signing in, copy the final address and try the Facebook video downloader. If Facebook displays a login wall, private-group page, unavailable message, or age gate, stop there.
Short fb.watch links can redirect to a full address. Let the redirect finish before copying again. This helps with link normalization, but it does not change privacy.
If you uploaded the video
Use Facebook's own account tools or the original file first. You can review the audience selector for your upload, but do not make a sensitive post public solely to send it through another service. A temporary privacy change can expose the video beyond its intended audience.
What Pullvio will not request
Pullvio should not need your Facebook password, session cookie, browser export, private-group HTML, or another person's credentials. A tool that asks for those items is taking on a different and much riskier access model.
The practical answer is simple: obtain an authorized copy from the uploader or use the platform's owner controls. Public-link tools are not a workaround for Facebook's audience rules.