I have a new Mac Pro and a Mac Mini on the same network. Twice a message has appeared saying files are being copied from the Mac Pro to the Mac Mini when I did not initiate the copy and in fact had sharing features turned off. How can this happen?

Next time it happens, get a screen shot. The text of the message and any details of the files involved will be of interest.

There can be shared network caches available and used for app installs, and add-on apps may well copy files around.

Continuity will definitely allow links and resources to be available, but that doesn’t show as a file copy dialog.

Download and run and post the output of EtreCheck here. Open a new reply here, press the button that looks like a printed page to get a text input box large enough to paste the hardware and software configuration report here.

The Apple store page you’ve linked to doesn’t match the icon shown. There’s no (obvious) use of the old cheese-grater on the new cheese-grater page. This is either old Apple software and/or something that doesn’t appear very often, or it’s not Apple software. You’re probably going to end up removing a package at a time, and see when this disappears, if this isn’t something that somebody here recognizes.

Various candidates here, unfortunately. No good culprits, though.

See if removing the anti-malware clears this.

Four backup tools are worth a look, for what they’re copying around.

GPG has its issues, but those are with the security of its intended purpose.

Haven’t looked at the status of TrueCrypt.

Flash and Java security has been problematic over the years.

Skype has been updated since 2018.

The TrueCrypt project was shut down several years ago so you are using really outdated software. You should be able to use VeraCrypt instead which is a fork of TrueCrypt that is still being maintained and updated.

I find the "Copy" and "Pasting" to be interesting, but I don't how to identify the reason for it.

Since you have a VM, then maybe you need to investigate the VM. The easiest thing would be to disable the VM.

Try booting into Safe Mode to see if you still have the problem.

Is FileSharing enabled?

Is remote access of any kind enabled?

The Pasting from "Ray's Mac Pro"... sounds a lot like the macOS Universal Clipboard actions

https://support.apple.com/en-us/HT209460

  /Library/Extensions
    [Loaded]     SophosFileProtection.kext (9.9.4 - SDK 10.14)
    [Loaded]     SophosWebProtection.kext (9.9.4 - SDK 10.14)

Suggest removing the anti-virus

Where is your iDrive pointing? Is it to a cloud server or to your Mac Pro?

Where is Carbon Copy Cloner pointing? Do you have a schedule CCC backup, although CCC does not generally have Finder style messages.

CrashPlan used to allow backing up to your own server, but they discontinued that, so I assume your CrashPlan is backing up to CrashPlan servers which you are paying for. Also CrashPlan did not display Finder style copy messages.

What is the Paragon software doing? I'm used to seeing Paragon's name associated with Windows NTFS file system access, but I'm not seeing the kernel extension I normally see when accessing an NTFS file system. So I'm wondering if you are running some other product from Paragon.

What is MySQL doing? Are you running some kind of database app that uses the Mac Pro for storage?

What are you using MacFUSE for. It is generally associated with file systems not native to macOS, or with file systems accessed over the network via ftp, ssh or sftp.

Have Activity Monitor running sorted by cpu and see what process is using cpu during the phantom copy.

The icons shown are of a two-generations-back Mac Pro, which implies that this isn't Apple software.

Remove the VM, and see if the messages go away?