Are you looking for an extension for Opera browser that lets you open links from right-click menu in Private tab? Privatize is the extension you’re looking for. You can open a new Private tab or Private Window in Opera from its menu, for Private window there is Ctrl+Shift+ N shortcut is available. If you’ve stumbled upon a shopping site link which you want to open in private tab then instead of opening the private tab and pasting the link in browser, its much convenient to open that link from context menu isn’t it?, Privatize extension adds that feature to Opera.
Install the extension which adds “‘Open in Private tab” to context menu for the link which you hover over with your Mouse pointer. You can easily identify a private tab in tab bar of Opera as it has open lock icon. You can close that tab to exit from private browsing mode.
Looking for an extension like this for Firefox use Open in Private browsing mode.