Yes, but...
Ultimately, the two screens are shared - both screens showing a part of the total, extended desktop. You can choose to run one or both screens as full screen. In my own usage, it depends on what you are doing. Sometimes the full-screen function on one screen, limits what you can do on the other screen to some extent. An example that I run into quite often: I start a youtube video (through Safari), showing on projector. When I expand that video window to full screen for the projector, the main monitor goes black, except for Safari's menu bar, which remains in view. When I am done with that video, hit escape, and the two screens return to my normal setup.
I also use some specialist presentation software which takes over both displays. The projector image is automatically in full screen mode. The local monitor is the controller for the presentation. I have to be cautious about doing anything else, as I can easily drop out of the presentation app, and lose the presentation while it is in progress, so that app takes over both displays while it is in use - although I can do certain limited tasks on the local display.
So, in my experience, the two displays do not often operate independently, but that's because of my use, and the apps that I choose to run.
Your results ultimately depend on the apps that you use, and how it is all set up.
One final thought: If you have your main monitor (your laptop screen, probably) on the right, and you have your dock on the right. It will be on the main monitor even if you plug in your external display. Probably the Dock will always be on the right edge, even with an extended desktop. If you have the Dock on the left side of your main display, THEN plug in your second display (set up to be on the left side of your pair), then the Dock should disappear from the main display, to show on the left edge of the extended desktop, so the left edge of your external display. That's simple! Dock at the bottom MIGHT move from screen to screen when you start using full screen mode on one or the other display. I can guess that the Dock could move around, depending on what you are doing. You might experiment with that, mostly so you can predict what the Dock will do.
The top menu bar, should exist on the display that is designated in the Displays/Arrangement tab. You can drag the menu bar from one screen to the other in that Arrangement pane. That's how you can decide with of the two displays will be used as the main display by your system. (note: I THINK that Catalina now has changed how some of that works, so if you are using Catalina there might be other differences. I am not there yet with my two-display setup, so not sure about that...)
