My mouse started moving around my screen on its own

In addition to your valid concern that someone may have had remote access to that Mac, there is at least one additional reason to completely erase that Mac and reconfigure it from the ground up, and that's due to having apparently used "CleanMyMac" at some time.

Other concerns are evident:

/etc/hosts - Count: 1

Why?

Erase that Mac. Follow these instructions: Erase your Mac and reset it to factory settings - Apple Support.

When repopulating its contents from its Time Machine backup, choose a date prior to the intrusive event:

Backup:

Time Machine information is limited without Full Disk Access

We don't know what that date might be.

Please post the full Etrecheck report as follows:

Copy the report

and use the Additional Text button to paste the report in your reply.

Then we can examine the report and see if we can determine the cause of the problem.

I concur with the assessment from @ John Galt regarding " CleanMyMac " aka " BrickMyMac "

There also appears that some Outdated Software that has been migrated form previous installation on another computer

The mix of CMM and outdated software spells and go mix and may be the source of many issues including the Mouse Issue

Start over as previous suggested

This time, do your best not to Migrate Older software but to install the current versions

Factor in the Apple Silicon aspect, were the newer software is using Universal Binary Software. 

It contains Both Intel and M1/ M2 / M3 ARMs binary code and refer to (  Universal 2 ).

When the user installs a 64 bit Application that contains Both Intel and M1 / M2/ M3  ARMs binaries - depending on factors, the Operating System may or may not invoke Rosetta 2.

The report indicates the presence of an extraneous entry. It may or may not be meaningful, but read Fixing a hacked /etc/hosts file - Apple Community. It explains what the Hosts file is used for, and how to fix it if it has been altered.

There are no references to JavaScript, which is completely unrelated to Java. It's the latter that appears to be old. It also may or may not be meaningful, but like anything else you choose to install on a Mac it's up to you to keep it updated.

There are still a lot of extraneous and unnecessary system modifications present. The report states:

System modifications - There are a large number of system modifications running in the background.

Any one of them can cause unexpected behaviour. You need to ask yourself what you need, and what you don't.

CleanMyMac also appears to remain, and is still affecting that Mac to an unknown extent. It has been my experience that the most expedient solution to having used it (or similarly categorized junk) is to erase the affected Mac.

Log out Apple ID from that device and refer to the apple authorised retailer. Open iCloud on another device check spam email or app that have not been using by you. Deleted it or uninstall it. Check the warranty for device support. Claim the new device if possible. Your imei device likely have been exposed

jerik19 wrote:

Basically just that. While I was working this afternoon, my mouse started moving around on its own like a user had remotely accessed my desktop. I immediately shutdown the device, restarted and disabled wifi and then went about trying to figure out how this happened. I'm running an M1 Pro 16' Macbook.

Screen sharing and all other sharing settings were off. The "allow access" box had "Administrators" listed, which I removed. There is only one user account associated with this device. I ran a virus scan that found nothing. I turned Apples trusted app filter back on. There is no evidence that anyone has accessed my icloud account but I did update my password. Oh, and I completed the security updates for MacOs that I had been putting of.

I noticed several weeks ago that I was getting a your "screen is being observed" notification while the device was locked and eventually isolated the problem to Safari but assumed this was just a bug from when I was screen sharing in the browser during a conference call earlier. Anyways, I think I've probably been hacked but not sure what my next steps should be to address the problem.

Third Party Security Software to " ran a virus scan "

The Mouse Jumping around uncontrollable could be associated with the Virus Scanner used on this computer.

There are no known Windows-like Viruses in the wild that self replicate and affect macOS, because of the underling UNIX  Foundation and Permission Limitation. 

The Operating System resides in a Sealed and Read Only Volume that cannot be opened by the User nor by Third Party Applications.

The only Entity that can open and modify or alter this Volume is Apple.

As for Hacking ?

This would have required the " Hacker " to have gained Physical and Direct Access to this computer.

They would have had to somehow, either bypassed your Strong Password for the Admin account or to have known / guessed the password.

Thereafter, to have installed their " Hacking " Method or Software to take full control of this computer

If you post the Etrecheck Report , yes will have a look at it

If the Mac is performing well and no longer exhibits any strange behavior, then yes I think you did everything right. You're all set.

The same thing happened to me! Did a red dot appear on your screen and start controlling it?