how to monitor desktop with Apple Remote desktop without needing desktop user approval?

On your client/end user machines try checking your Remote Login and Remote Management settings in:  > System Preferences > Sharing

Remote Login - ensure that at least "Allow access for: Only these users: Administrators" is enabled

That way other non-admin accounts cannot remote in with a shared password, etc.

Remote Management - ensure that "All access for: All Users" is enabled AND the options button will be selectable. Click on Options... then hold the option key and click on one of the check boxes OR select the ones you'd like allowed for the admin mac to be able to see/function on client systems.

OR you can select "Allow access for: Only These Users:" and select the local admin account and then click Options, that way it grants the privileges for admins only versus any user who has a password to the machines.

Can you provide a screen shot of your Mac sharing pane?

Screen sharing preference should be unchecked as Remote Login and Management have redundant processes.

This is how they should appear:

1) Remote Login must be enabled - Select either All Users or Only These Users and select the Administrator

2) Remote Management also must be enabled - Select either All Users or Only These Users and select an Administrator then follow Step 3

3) Click on Options... and option click one of the check boxes to simultaneously select all options or individually click each option you need/require

try restarting Apple Remote Desktop and your admin Mac. when Apple Remote Desktop launches click File > Refresh

and attempt to connect to the devices again.