The correct behaviour when installing NoMachine on Catalina can be seen below. Some of the issues encountered when using NoMachine v. 6.8.1 or earlier are outlined in the article available here: https://www.nomachine.com/AR10Q01057.
It's important to update the NoMachine installation to the latest version in order to solve such issues. Ref. https://www.nomachine.com/softwareupdates
The expected behavior is the following.
When installing NoMachine v. 6.8.2 or later on Catalina a system dialog is issued for the Accessibility Access (Events). If you're administrator, grant access to 'nxnode' application by clicking on 'Open System Preferences':
Allow 'nxnode' (tick the box):
Then add 'NoMachine' to Screen Recording: a system dialog is issued to grant access to 'NoMachine' to screen recording functionalities. Click on 'Open System Preferences':
and allow 'NoMachine' to record the content of your screen:
The next step is to allow file-access permission to NoMachine. A system dialog request to grant permission will be displayed whenever NoMachine needs access to a file or folder.
Launch the NoMachine application. The system should issue a dialog asking to grant permissions to NoMachine to access files e.g. on your Desktop or Documents folder. Grant permission to NoMachine from such dialog:
Click on 'OK' and add NoMachine to the list of allowed applications for Full Disk Access:
It's also possible to grant 'NoMachine' with 'Full Disk Access permission' by editing the system settings: Open Settings -> Security & Privacy -> Privacy group and select Full Disk Access permission. Add NoMachine to the list of allowed applications for Full Disk Access.
When 'grab the mouse input' is selected in the NoMachine client preferences:
the system issues a dialog asking to grant NoMachine permission to control this computer:
Click on Open System Preferences and add 'NoMachine'.