net.duplicati.server.plist

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
    <key>Label</key>
    <string>net.duplicati.server</string>
    <key>ProgramArguments</key>
    <array>
        <string>/Applications/Duplicati.app/Contents/MacOS/duplicati-server</string>
        <string>--webservice-port=8200</string>
    </array>
    <key>RunAtLoad</key>
    <true/>
    <key>LaunchOnlyOnce</key>    
    <true/>
    <key>KeepAlive</key>
    <true/>
</dict>
</plist>

net.duplicati.tray-icon.plist

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
    <key>Label</key>
    <string>net.duplicati.tray-icon</string>
    <key>ProgramArguments</key>
    <array>
        <string>open</string>
        <string>/Applications/Duplicati.app/</string>
        <string>--args</string>
        <string>--hosturl="http://localhost:8200"</string>
        <string>--no-hosted-server</string>
    </array>
    <key>RunAtLoad</key>
    <true/>
</dict>
</plist>

Download and copy Duplicate to your /Applications folder. Don't run it yet.

Create the two files from the above:

/Library/LaunchDaemons/net.duplicati.server.plist
/Library/LaunchAgents/net.duplicati.tray-icon.plist

And then run Duplicati

launchctl load -w /Library/LaunchDaemons/net.duplicati.server.plist
launchctl load -w /Library/LaunchAgents/net.duplicati.tray-icon.plist

Running on MacOS Catalina (10.15.3) with Duplicati (2.0.5.1_beta_2020-01-18).