In addition to the reasons other have suggested, one reason that this is not done client side is that the PhotoDNA tech that is used for the fingerprinting is not something that Facebook can share or make available. It's primary use is in matching posted images with known databases of child porn images, so providing a mechanism for someone to reverse engineer the system and develop a means of bypassing it is not going to happen.

They could just send fingerprints, sure.

The problem is that you presumably want the image to be blocked immediately upon upload.

But facebook is worried that people will abuse the system, and they don't want images to be instantly wrongfully blocked until someone can review it.

Abuse-resistance. Instant blocking. Perfect privacy. Pick two. There's no perfect solution.

Why not suspend reported images for review. If the review indicates the reporter is abusing the system, blacklist the reporter from the automatic function. Also block new and recent accounts, and stop automatic blocking for images on frequently targeted accounts and major accounts (e.g. If someone is reporting Donald Trump, manually check, don't automatically remove).

Doesn't this system already exist? But reporting an image means it wasn't preemptively blocked, the benefit of the new system.

If you can send only the fingerprint, what stops you from uploading photos you are not in?

A better idea may be an uploading tool where you can obscure some parts for verification.

That sounds like a solution to me, it satisfies whatever the need is for human involvement while enabling the user to redact the weaponizable elements of the picture.

That's an interesting idea, but might break whatever hashing method is used?

The idea is to hash the original image, but to send the obscured one for human verification. However this requires a different hashing method (such a Merkle tree) to know the unobscured parts match.

Seems like anybody can upload any body.