I'm not the tech at Verizon but my guess is they transfer the contacts info but not the media files.
And even if they did, the transfer machine they have just shows whether the transfer is done or not, and files names. I don't believe they have the image reader to see the pictures and videos.
But, it's possible that when they do the transfer, they may temporarily store the data before transfering it to the new phone and who knows what they do with it. Most likely delete but I don't know.
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