AirDrop: do both computers need to be running the same iOS

AirDrop noob here: I'm having difficulty moving files from my desktop to the laptop using airdrop. Both have bluetooth active, so I am wondering if the problem is the desktop is running Big Sur and the laptop is running Catalina.

The difference between the two has caused some iCloud issues, which is why I am not using it in this situation (Also, I want to preserve the filing system I have in place). So, I'm wondering if that same problem is the reason I can't get AirDrop to work. That or I'm just dum.

I have been following the steps listed below. Also, I have noticed on my desktop (Big Sur), the laptop shows up in the AirDrop Finder window; however, the reverse is not true.

Use AirDrop on your Mac to send files to devices near you

Thanks in advance for your advice.