Could not establish connection with Unity package manager local server process - Unity?

1. Press the Windows button and search “configure proxy server”.
2. In the ‘Connections’ tab. Press on LAN setting.
3. That will land you to this window below and set it to the same setting as you are seeing in the picture Settings Image 1

Also, do the same in PC Setting if you are using Windows 8.1 or 10

1. Go to PC Setting
2. Click on ‘Network’
3. Click on ‘Proxy’
4. Choose the same setting as you are seeing in the picture below Settings Image 2

Now

1. Right-click on ‘My Computer’ icon
2. Go to properties
3. Click on ‘Advanced System Settings’
4. Click on ‘Environment Variable’
5. Delete any HTTP_PROXY or HTTPS_PROXY entry from both the panel(User and System). They both are not important for the network.

[Edited to improve] If still Asset store is not working and asking for checking HTTP_PROXY and HTTPS_PROXY settings

1. Go to Windows Firewall in Control Panel
2. Click on 'Allow a Windows app or feature through Windows Firewall'
3. Allow Unity's software in Private and Public Section(tick on them)
4. Click 'OK'