Your desires can be met, but not with the physical setup you describe.
if you can get both computers onto the Internet at the same time, without re-cabling, you already have a highway between them. It runs through the Switch function of your Router. You can enable File Sharing in
System Preferences > Sharing > File Sharing
By default, this will allow you to use a Shared folder on each Mac to drag and drop files onto the other Mac's drive. With more advanced settings in this pane, you can specify additional folders to Share. "Share" in this context is a one-way connection -- there is no implicit reciprocity.
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If, instead of a Drive, you used a Network-Attached Storage device (NAS) which has its own processor, you could indeed connect this drive over a "private" Ethernet cable, and also connect it to you other Mac over a similar connection. i am not conviced the increased complexity is worth the effort.
I use a MacOS Server (separate computer) mostly as a File server on my Mome Network, and connect via Gigabit Ethernet using Gigabit Ethernet Switches. My Internet Router is on a spur off this network.