OK, here goes. I'm on Linux; location of the settings menu may differ, depending on what you use.
- Go to Tools menu, select "Add-Ons and Themes". You can also go to about:addons in the URL bar.
- On that page, I have on the left: Recommendations, Extensions, Themes, Plugins. Choose Extensions.
- You should be in a page that says, "Manage Your Extensions" near the top. Scroll down to find the one in question, Sidebery in this case.
- To the right of the enable/disable toggle, there is an icon with three dots.
- Click the three-dot icon, select "Manage".
- One of the options is, "Run in Private Windows". Select "Allow".
If I open a private window, Sidebery isn't open by default, but is available if the extension icon is clicked in the bar under the tabs, or I click Ctrl-e to toggle back and forth.